Initial data sets with zero mass are contained in plane wave spacetimes

We combine spinorial and level-set methods to show that asymptotically flat initial data sets, with E=|P| and satisfying the dominant energy condition, can be isometrically embedded into a plane wave spacetime. This completely settles the rigidity of the spacetime positive mass theorem for spin manifolds in all dimensions. Additionally, a non-vacuum example with the borderline decay rate is constructed. This is a joint work with Sven Hirsch.