  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 modelling 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.
