The Full Penrose Inequality, the Hoop Conjecture, and Quasi-Local Mass

1973年，R.Penrose提出了一个猜想，它通过一个不等式将时空的总质量与其事件视界(黑洞边界)的面积联系起来：总质量的平方大于或等于事件视界的总面积除以16pi。本研究的主要目的是证明完整的彭罗斯不等式，并用所发展的方法来研究广义相对论中与质量有关的几个问题。最好将这一猜想看作是时空的任意类空片的一个不等式，并且已经在该片具有零秒基本形式(时间对称情况)的情况下得到了证实(对于一个黑洞，由Huisken和Ilmanen，对于有限多个黑洞，由Bray)。鉴于H.Bray和作者最近的一项发现，对一般切片的猜想似乎第一次近在咫尺，该发现将问题归结为求解一个规范的偏微分方程组。出乎意料的是，这一新方法揭示了彭罗斯不等式、环状猜想和刘友准局域质量之间的显著联系。作者打算进一步研究和发展这些联系，目的是得到一个明确的黑洞形成的充要条件，以及克服其固有困难的刘游质量的修正版本。彭罗斯不等式最初是由彭罗斯提出的，是为了研究今天经典广义相对论中最重要的公开问题，即宇宙审查猜想。这一猜想断言，每当时空演化中出现奇点(预计这是一种普遍现象)时，它们一定总是被事件视界隐藏在外部世界，也就是说，它们总是位于黑洞内部。根据彭罗斯的启发式推导，彭罗斯不等式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有效性的普遍信念，这反过来又是决定广义相对论作为物理理论表现得有多好的根本。此外，对Liu-Yau准局域质量的修正可能会导致引力场的局域能量密度的第一个完全有意义的表达式。这反过来应该导致爱因斯坦方程柯西问题的研究以及黑洞形成(环状猜想)的研究取得新的进展。最后，广义相对论的一个普遍主题是，涉及爱因斯坦方程(如彭罗斯不等式和正质量定理)的初始数据集的定理通常首先在较容易的时间对称情况下得到证明，而一般情况下则是一些如何还原到时间对称的情况。预计为完全彭罗斯不等式开发的方法将提供一种新的强有力的工具来实现时间对称性的这种约化，因此将在广义相对论的广泛问题上有许多应用。

In 1973 R. Penrose proposed a conjecture which 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. The primary goal of this research is to prove the full Penrose Inequality, and to use the methods developed to study several related problems concerning mass in General Relativity. It is best to view this conjecture as an inequality for an arbitrary spacelike slice of a spacetime, and it has been confirmed (by Huisken and Ilmanen for one black hole, and by Bray for finitely many black holes) in the case that the slice has zero second fundamental form (the time symmetric case). For the first time the conjecture for a general slice appears to be within reach in light of a recent discovery by H. Bray and the author, which reduces the problem to solving a canonical system of partial differential equations. Unexpectedly, this new method has revealed a significant connection between the Penrose Inequality, the Hoop Conjecture, and the Liu-Yau quasilocal mass. The author intends to further investigate and develop these connections, with the aim of obtaining a definitive necessary and sufficient condition for black hole formation, as well as a modified version of the Liu-Yau mass which overcomes some of its inherent difficulties. The Penrose Inequality was originally put forth by Penrose to study the most important open question in classical General Relativity today, namely the Cosmic Censorship Conjecture. This conjecture asserts that whenever singularities occur in the evolution of spacetime (which is expected to be a generic phenomenon) they must always be hidden from the outside world by an event horizon, that is, they must always lie inside a black hole. According to Penrose's heuristic derivation, 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, which in turn is fundamental for determining how well General Relativity is behaved as a physical theory. Furthermore, a correction to the Liu-Yau quasilocal mass would possibly lead to the first fully meaningful expression of local energy density for the gravitational field. This in turn should lead to new advances in the study of the Cauchy problem for the Einstein Equations, as well as in the study of black hole formation (the Hoop Conjecture). Lastly, it is a general theme in General Relativity that theorems involving initial data sets for the Einstein Equations (such as the Penrose Inequality and Positive Mass Theorem) are often proved first in the easier time symmetric case, while the general case is then some how reduced back to time symmetry. It is expected that the methods developed for the full Penrose Inequality will provide a new powerful tool for making such a reduction to time symmetry, and will therefore have numerous applications to a wide range of problems in General Relativity.