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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).
Important caveats remain. The theorems assume the dominant energy condition µ ≥ |J| and certain technical decay and regularity conditions at infinity (the authors impose stronger differentiability assumptions than are sometimes assumed in the literature to make earlier results directly applicable). The rigidity statement — the precise description of the equality case — is still open in the non-spin setting when the extrinsic curvature is not the metric. The asymptotically anti-de Sitter (AdS) version of the spacetime positive mass theorem is also still unknown in full generality without spin assumptions. The authors note related recent work that proves similar inequalities under different additional positivity hypotheses.