Mass 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命题：静态延拓猜想、完全彭罗斯不等式和箍猜想。其中第一个来自一个特别有前途的定义准局部质量由于巴特尼克，它试图本地化的总（或ADM）的质量。虽然这个定义满足大多数期望的属性，因此有可能是非常有用的，其抽象和非建设性的性质产生严重的限制理解和适用性。为了纠正这个问题的静态扩展猜想断言，准局部质量可以计算为静态真空爱因斯坦方程满足一定的几何边界条件的解决方案的ADM质量。在作者与M.安德森，这个边值问题的存在性已经建立在一个合理的非退化假设下，对给定的边界数据，这一发展已经把猜测触手可及。彭罗斯不等式通过不等式将时空的总质量与视界（黑洞边界）的面积联系起来：总质量的平方大于或等于视界的总面积除以16 π。这可以被看作是对时空中任意类空切片的一种猜想，在这种情况下，一个重要的特殊情况（当切片的第二基本形式消失时）已经被Huisken和Ilmanen（一个黑洞）和Bray（许多黑洞）独立地证实了。在最近与布雷的合作中，作者成功地将这个问题简化为求解一个典型的偏微分方程组，并证实了在特殊情况下存在。这个项目的主要目标是通过证明一个普遍存在的结果来完成这个程序，从而建立彭罗斯猜想。虽然在广义相对论中有许多计算有界区域的引力加物质能量含量（所谓的准局域质量）的方法，但似乎都受到这样或那样的困扰。在巴特尼克质量的例子中，唯一的问题是它的定义的抽象性。如果静态可拓猜想得到验证，这个唯一的困难应该得到解决，从而为几个应用开辟了道路。例如，一般认为，只要足够的质量集中在一个足够小的区域，引力坍缩必然随之而来，并导致黑洞-这种说法通常被称为箍猜想。虽然有许多方法可以精确地描述区域的大小，但在建立准局域质量的适当概念之前，这个猜想的严格和完整版本仍然是难以捉摸的。另一个有希望的应用适当定义的概念的准局部质量，是长时间存在的爱因斯坦方程。由于这些方程形成了一个双曲系统（在固定规范之后），人们很想寻找一种类似于标量波动方程经典理论的基于能量方法的理论。实现这种方法的一个主要障碍是缺乏适当的引力场能量概念（准局部质量的一个组成部分）。此外，与静态扩张猜想相关的边值问题实际上是Ricci算子的椭圆边值问题。因此，开发的技术来研究这个猜想应该证明是有用的，当调查适定的边值问题在其他几个设置，例如，里奇流流形上的边界。至于彭罗斯不等式，它最初是由彭罗斯提出来研究今天广义相对论中最重要的开放问题，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，彭罗斯不等式本质上是宇宙审查成立的必要条件。因此，如果彭罗斯不等式得到证实，它将大大增加对宇宙监督有效性的普遍信念。最后，我们的方法开发的完整的彭罗斯不等式，预计将提供一个新的强大的工具，减少一般的初始数据集的情况下的时间对称的问题，因此将有许多应用到广泛的其他问题在广义相对论。

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.