Proof of a Null Geometry Penrose Conjecture

摘要

Beginning in 1973, Roger Penrose wondered about the relation between the mass of the universe and the contributions of black holes ([1], [2]). All we can see are their outer boundaries or "horizons," the size of which should determine their mass contributions. He conjectured that the total mass of a spacetime containing black hole horizons with combined total area |Σ| should be at least √|Σ|/16π. On the one hand, this conjecture is important for physics and our understanding of black holes. On the other hand, Penrose's physical arguments lead to a fascinating conjecture about the geometry of hypersurfaces in spacetimes. For a spacelike "Riemannian" slice ("the universe at a particular time") with zero curvature (zero second fundamental form) the conjecture is known as the Riemannian Penrose Inequality and was first proved by Huisken-Ilmanen in 2001 (for one black hole) and then by the first author shortly thereafter, using two different geometric flow techniques. This article concerns a formulation of the conjecture for certain light-cone-like "null" hypersurfaces in spacetimes called the Null Penrose Conjecture (NPC). Over the last ten years, there has been a great deal of progress on the NPC, culminating in a recent proof of the conjecture in a fair amount of generality for smooth null cones by the second author. One surprising fact is that these null hypersurfaces, under physically inspired curvature conditions on the spacetime, have "monotonic" properties (increasing as expected), including cross sectional area, notions of energy, and, as we'll see with Theorem 1 below, a new notion of mass.