Proof of a Null Penrose Conjecture Using a New Quasi-local Mass.

In the theory of general relativity, the Penrose conjecture claims a lower bound for the mass of a spacetime in terms of the area of an outermost horizon, if one exists. In physical terms, this conjecture is a geometric formulation of the statement that the total mass of a spacetime is at least the mass of any black holes that are present, assuming non-negative energy density. For the geometry of light-rays emanating off of a black hole horizon (called a nullcone) the Penrose conjecture can be reformulated to the so-called Null Penrose Conjecture (NPC). In this thesis, we define an explicit quasi-local mass functional that is non-decreasing along all foliations (satisfying a convexity assumption) of nullcones. We use this new functional to prove the NPC under fairly generic conditions.