  We provide a new proof of the equality case of the spacetime positive mass
theorem, which states that if a complete asymptotically flat initial data set
$(M, g, k)$ satisfying the dominant energy condition has null ADM
energy-momentum (that is, $|E|=|P|$), then $(M,g)$ must isometrically embed
into Minkowski space with $k$ as its second fundamental form. Previous proofs
either used spinor methods [Wit 81, BC96, CM06], relied on the Jang equation
[HL20, Eic13], or assumed three spatial dimensions [HZ22]. In contrast, our new
proof only requires knowing that $E\ge|P|$ for all complete initial data sets
near $(g,k)$ on $M$ satisfying the dominant energy condition.
