The spacetime positive mass theorem in dimensions less than eight

Details The spacetime positive mass theorem in dimensions less than eight The spacetime positive mass theorem in dimensions less than eight

2011-10-10

arxiv.org/abs/1110.2087v1

Mathematics

Differential Geometry

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Scientific paper

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 inequality $E\ge |P|$ holds, where $(E,P)$ is the ADM energy-momentum vector of the initial data set. 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 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 author. Since MOTS do not arise from a variational principle, an important part of our proof to introduce an appropriate substitute for the area functional used in the time-symmetric case. As part of our proof, we 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. A refinement of the density argument allows us to approximate any given data set by one which is identical with a Kerr slice outside a compact set preserving the dominant energy condition. This enables us to give an alternative proof of the main theorem by reducing it to the positive energy theorem.

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