  We give a lower bound for the Lorentz length of the ADM energy-momentum
vector (ADM mass) of 3-dimensional asymptotically flat initial data sets for
the Einstein equations. The bound is given in terms of linear growth `spacetime
harmonic functions' in addition to the energy-momentum density of matter
fields, and is valid regardless of whether the dominant energy condition holds
or whether the data possess a boundary. A corollary of this result is a new
proof of the spacetime positive mass theorem for complete initial data or those
with weakly trapped surface boundary, and includes the rigidity statement which
asserts that the mass vanishes if and only if the data arise from Minkowski
space. The proof has some analogy with both the Witten spinorial approach as
well as the marginally outer trapped surface (MOTS) method of Eichmair, Huang,
Lee, and Schoen. Furthermore, this paper generalizes the harmonic level set
technique used in the Riemannian case by Bray, Stern, and the second and third
authors, albeit with a different class of level sets. Thus, even in the
time-symmetric (Riemannian) case a new inequality is achieved.
