The spacetime positive mass theorem in dimensions less than eight

TL;DR

The paper proves the spacetime positive mass theorem \(E \ge |P|\) for asymptotically flat initial data satisfying the dominant energy condition in dimensions \(3 \le n < 8\), without a spin assumption. The proof is by contradiction and induction, using marginally outer trapped hypersurfaces and a density theorem reducing to harmonic asymptotics with strict dominant energy condition.

Key findings

• Main theorem: for \(3 \le n < 8\), any \(n\)-dimensional asymptotically flat initial data set \((M,g,k)\) satisfying the dominant energy condition has ADM energy-momentum vector \((E,P)\) satisfying \(E \ge |P|\).
• The proof is self-contained rather than by reduction to a known case; it gives a new proof of the \(E \ge 0\) theorem for non-time-symmetric data.
• The \(n=3\) case is completed by contradiction to the Gauss-Bonnet theorem; for \(3 < n < 8\), the proof is by induction and contradicts the time-symmetric positive mass theorem in dimension \(n-1\).
• A density theorem shows that an asymptotically flat initial data set satisfying the dominant energy condition can be perturbed to one with harmonic asymptotics and strict dominant energy condition \(\mu > |J|\), with small changes in \(g\), \(\pi\), \(E\), and \(P\).

Methods

• Proof by contradiction and induction on dimension. Assuming \(E < |P|\), the authors use the density theorem to assume harmonic asymptotics and strict dominant energy condition, with \(P\) pointing in the vertical direction.
• They construct stable marginally outer trapped hypersurfaces (MOTS) \(\Sigma_{\rho,h}\) in vertical cylinders \(C_\rho\) between barrier planes, then extract a complete MOTS limit \(\Sigma_\infty\) as \(\rho \to \infty\).
• For \(n=3\), stability and Gauss-Bonnet give a contradiction. For \(3 < n < 8\), they introduce a functional \(F\), pick a height \(h_\rho\) and vector field \(X\) satisfying \(D\theta^+|_{\Sigma_{\rho,h_\rho}}(X)=0\) and \(DF|_{\Sigma_{\rho,h_\rho}}(X)\ge 0\), derive a stability-like inequality, and use a conformal change to produce an asymptotically flat scalar-flat metric of negative energy, contradicting the time-symmetric theorem in dimension \(n-1\).
• The theory of \(\lambda\)-minimizing currents substitutes for a variational principle for MOTS; the assumption \(n<8\) ensures the relevant MOTS are smooth.

Limitations

• The paper states that the theorem does not include a characterization of the equality case \(E=|P|\); the natural conjecture is that \(E=|P|=0\) and \((M,g)\) embeds isometrically into Minkowski space with second fundamental form \(k\), but the proof by contradiction means the equality case requires a substantially new idea.
• The paper states that the restriction \(n<8\) is used only for regularity of the MOTS; for \(n>8\), the lemma still holds except that the \(\lambda\)-minimizing boundaries are only regular away from a thin singular set.
• Evident from the text: the method is non-constructive for the equality case and relies on perturbation to harmonic asymptotics and strict dominant energy condition, so it does not directly address rigidity or higher-dimensional singularities.