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.