We consider localized deformation for initial data sets of the Einstein field equations with the dominant energy condition. Deformation results with the weak inequality need to be handled delicately. We introduce a modified constraint operator to absorb the first order change of the metric in the dominant energy condition. By establishing the local surjectivity theorem, we can promote the dominant energy condition to the strict inequality by compactly supported variations and obtain new gluing results with the dominant energy condition. The proof of local surjectivity is a modification of the earlier work for the usual constraint map by the first named author and R. Schoen and by P. Chru\'sciel and E. Delay, with some refined analysis.
We prove the spacetime positive mass theorem in dimensions less than eight. This theorem
asserts that for any asymptotically flat initial data set that satisfies the dominant energy condition,
the inequality E>=|P| holds, where (E, P) is the ADM energy-momentum vector. Previously,
this theorem was only known for spin manifolds . Our approach 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 of this theorem [30, 27]. 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 that is used in the time-symmetric case to single out certain minimal hypersurfaces.
We also establish a density theorem of independent interest and use it to reduce the general case of
the spacetime positive mass theorem to the special case of initial data that has harmonic asymptotics
and satisfies the strict dominant energy condition.