The Spacetime Penrose Inequality: Conditional Results for Stable MOTS and General Trapped Surfaces

Abstract

We present a rigorous proof of the Spacetime Penrose Inequality relating the ADM mass to the area of trapped surfaces in asymptotically flat initial data sets satisfying the dominant energy condition. The main theorem establishes that the ADM mass is bounded below by the square root of the area divided by 16 pi for an area-maximizing marginally outer trapped surface (MOTS), subject to a distributional favorable jump condition which we prove is structurally guaranteed by KKT optimality. The extension to the outermost MOTS remains conditional on the hypothesis that the area maximizer coincides with the outermost MOTS, or equivalently on Weak Cosmic Censorship. We explicitly flag that without this condition, the proof for general trapped surfaces does not go through, as evidenced by binary merger counterexamples. We provide a complete double-limit analysis of the Agostiniani-Mazzieri-Oronzio level-set flow on the singular Jang space, resolving regularity and boundary-term obstructions. In the equality case, the initial data embed isometrically into the Schwarzschild spacetime.