Equality in the spacetime positive mass theorem II

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.