  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.
