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

1973年R.彭罗斯提出了一个猜想，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。 本研究的主要目的是证明完整的彭罗斯不等式，并利用所发展的方法研究广义相对论中与质量有关的几个问题。最好把这个猜想看作是时空中任意类空切片的不等式，并且在切片的第二基本形式为零的情况下（时间对称的情况），这个猜想已经得到了证实（Huisken和Ilmanen对一个黑洞，Bray对许多黑洞）。 根据H。布雷和作者，这减少了问题，以解决一个典型的偏微分方程组。 出乎意料的是，这种新方法揭示了彭罗斯不等式，箍猜想和刘丘准局部质量之间的重要联系。 作者打算进一步研究和发展这些联系，目的是获得黑洞形成的一个确定的必要和充分条件，以及克服其固有困难的Liu-Yau质量的修改版本。 彭罗斯不等式最初是彭罗斯为了研究当今经典广义相对论中最重要的未解决的问题，即宇宙审查猜想而提出的。 这一猜想断言，无论何时在时空演化中出现奇点（这被认为是一种普遍现象），它们必须总是被事件视界从外部世界隐藏起来，也就是说，它们必须总是位于黑洞内部。 根据彭罗斯的启发式推导，彭罗斯不等式本质上是宇宙监督成立的必要条件。因此，如果彭罗斯不等式被证实，它将大大增加对宇宙监督有效性的普遍信念，而这反过来又是确定广义相对论作为物理理论表现如何的基础。 此外，对Liu-Yau准定域质量的修正可能会导致第一个完全有意义的引力场局域能量密度的表达式。 这反过来又会导致爱因斯坦方程的柯西问题的研究取得新的进展，以及黑洞形成（箍猜想）的研究。最后，广义相对论中的一个普遍主题是，涉及爱因斯坦方程初始数据集的定理（如彭罗斯不等式和正质量定理）通常首先在更容易的时间对称情况下得到证明，而一般情况则在某种程度上还原到时间对称。 预计为完整的彭罗斯不等式开发的方法将提供一个新的强有力的工具，使这种减少时间对称性，因此将有许多应用到广义相对论的广泛问题。

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.