The spacetime positive mass theorem in dimensions less than eight

TL;DR

This paper proves the spacetime positive mass theorem for dimensions less than eight using a novel approach involving marginally outer trapped hypersurfaces, extending previous results beyond spin manifolds.

Contribution

It introduces a new proof of the positive mass theorem in low dimensions without the spin assumption, utilizing MOTS and a modified area functional.

Findings

Validated the positive mass inequality E ≥ |P| for non-spin manifolds in <8 dimensions.

Developed a density theorem to reduce general cases to harmonic asymptotics.

Extended minimal hypersurface techniques to MOTS in the proof.

Abstract

We prove the spacetime positive mass theorem in dimensions less than eight. This theorem states that for any asymptotically flat initial data set satisfying the dominant energy condition, the ADM energy-momentum vector $(E,P)$ of the initial data satisfies the inequality $E \ge |P|$. Previously, this theorem was proven only for spin manifolds by E. Witten. Our proof is a modification of the minimal hypersurface technique that was used by the last named author and S.-T. Yau to establish the time-symmetric case. Instead of minimal hypersurfaces, we use marginally outer trapped hypersurfaces (MOTS) whose existence is guaranteed by earlier work of the first named author. An important part of our proof is to introduce an appropriate substitute for the area functional used in the time-symmetric case. We also establish a density theorem of independent interest that allows us to reduce the general case of our theorem to the case of initial data that has harmonic asymptotics and satisfies the strict dominant energy condition.