A team of mathematicians proves that, under a standard physical energy condition, the total energy of a space-like slice of spacetime is at least as large as the length of its total momentum in every space dimension at least three. In symbols this is the inequality E ≥ |P|, where E is the total energy and P is the total linear momentum. The result covers two physically relevant kinds of initial data: those that look flat far away (“asymptotically flat”) and those that approach a hyperbolic slice of spacetime at infinity (“asymptotically hyperboloidal”).

To get this conclusion the authors build on a recent breakthrough by Brendle and Wang about the purely spatial (Riemannian) positive mass theorem. They reduce the spacetime statement to that Riemannian case by solving a geometric partial differential equation called the Jang equation. Because the raw Jang equation can develop singularities, they work with a capillarity-regularized version and extract a subsequential limit, called a Jang graph, where they can apply the Riemannian theorem.

A central technical advance in the paper is a new regularity result for these potentially singular Jang graphs. The authors show each limiting Jang graph is “almost minimizing” in a precise sense and that its singular set is small: its Hausdorff dimension is at most n−7. In low dimensions (n ≤ 7) this implies the Jang graph is smooth. They also address possible blow-ups of the Jang solution near certain trapped surfaces and avoid accumulation of singularities by a conformal blow-up argument done piece by piece.

The paper states two main theorems. For asymptotically hyperboloidal initial data that satisfy the dominant energy condition (energy density μ at least the magnitude of the momentum density J), the total energy–momentum vector satisfies E ≥ |P|. If the spatial manifold is spin or if the extrinsic curvature equals the metric (a special case), equality occurs exactly when the data embed isometrically as a spacelike slice of Minkowski space. For asymptotically flat data the analogous inequality holds for the ADM energy and momentum, and equality is characterized by embedding into a special class of gravitational wave spacetimes called pp-waves (which reduce to Minkowski space in low dimensions).