  The rigidity of the spacetime positive mass theorem states that an initial
data set $(M,g,k)$ satisfying the dominant energy condition with vanishing mass
can be isometrically embedded into Minkowski space. This has been established
by Beig-Chru\'sciel and Huang-Lee under additional decay assumptions for the
energy and momentum densities $\mu$ and $J$. In this note we give a new and
elementary proof in dimension 3 which removes these additional decay
assumptions. Our argument uses spacetime harmonic functions and Liouville's
theorem. We also provide an alternative proof based on the Killing development
of $(M,g,k)$.

Differential Geometry;General Relativity and Quantum Cosmology;Analysis of PDEs