Initial data sets with vanishing mass are contained in pp-wave spacetimes

In 1981, Schoen-Yau and Witten showed that in General Relativity both the total energy $E$ and the total mass $m$ of an initial data set modeling an isolated gravitational system are non-negative. Moreover, if $E=0$, the initial data set must be contained in Minkowski space. In this paper, we show that if $m=0$, i.e. if $E$ equals the total momentum $|P|$, the initial data set must be contained in a pp-wave spacetime. Our proof combines spinorial methods with spacetime harmonic functions and works in all dimensions. Additionally, we find the decay rate threshold where the embedding has to be within Minkowski space and construct non-vacuum initial data sets with $m=0$ in the borderline case. As a consequence, this completely settles the rigidity of the spacetime positive mass theorem for spin manifolds.