The Case of Equality for the Spacetime Positive Mass Theorem

Abstract

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–Chruściel 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).