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

TL;DR

This work proves a spacetime Penrose inequality for asymptotically flat initial data under the dominant energy condition, linking ADM mass to the area of trapped surfaces. It builds a four-stage pipeline—Generalized Jang reduction, conformal sealing, corner smoothing, and the Agostiniani–Mazzieri–Oronzio (AMO) p-harmonic level-set flow—with a careful double limit as p\to 1^+ and smoothing parameter \varepsilon\to 0 to establish area–mass monotonicity even in low-regularity geometries. A key novelty is replacing pointwise favorable trace conditions with a distributional Karush–Kuhn–Tucker (KKT) condition, ensuring the correct sign in the boundary term for the AMO monotonicity; equality cases embed into Schwarzschild spacetime. The results are sharp for outermost MOTS under the favorable jump or Weak Cosmic Censorship, and extend to general trapped surfaces under compactness or cosmic censorship assumptions, with a quantitative extension when DEC is violated. The framework synthesizes PDE techniques on Lipschitz manifolds, capacity removability of singularities, and geometric measure theory to achieve a rigorous, sharp Penrose inequality in the spacetime setting, advancing both mathematical relativity and geometric analysis.

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.

Figures (11)

• Figure 1: Logical flow of the proof. Geometric constructions progress along the top row, while the lower row records the analytic invariants that authorize each passage.
• Figure 2: Proof dependency graph.
• Figure 3: The geometric action of the Generalized Jang Equation. The graph function $f$ blows up at the marginal surface $\Sigma$ in the initial data (left), creating a manifold $\overline{M}$ (right) with a new cylindrical end $\mathcal{E}_{cyl}$ where the scalar curvature condition becomes favorable.
• Figure 4: Smoothing the internal corner. The singular interface $\Sigma$ is replaced by a smooth collar $N_{2\epsilon}$. The curvature "dip" inside the collar is controlled by the $L^{3/2}$ estimate.
• Figure 5: The smoothing of the internal corner. The Lipschitz metric (left) has a mean curvature jump at $\Sigma$. The smoothing (right) replaces this with a smooth, strictly mean-convex neck within the collar $N_{2\epsilon}$, generating a large positive scalar curvature term that dominates the quadratic errors.

Theorems & Definitions (748)

• Remark 1.1
• Lemma 1.2: KKT interface
• Theorem A: Existence of area maximizer
• Theorem B: Penrose inequality for MOTS
• Theorem C: Extension to general trapped surfaces
• Remark 1.3
• Theorem D: Distributional favorable jump
• Theorem 2.1: Rigidity
• Theorem 2.2: Extended Inequality under DEC Violation
• Remark 2.3: Forward references