The case of equality for the spacetime positive mass theorem

TL;DR

This paper provides a new, elementary proof of the rigidity part of the spacetime positive mass theorem in three dimensions, removing previous decay assumptions and using harmonic functions and Killing development techniques.

Contribution

It introduces a simplified proof for the rigidity of the spacetime positive mass theorem without decay assumptions, expanding its applicability.

Findings

Proves rigidity of the spacetime positive mass theorem in 3D without decay assumptions.

Uses spacetime harmonic functions and Liouville's theorem for the proof.

Provides an alternative proof via Killing development.

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-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 $(M,g,k)$.