= Parallel-spinor classification of vacuum four-geometries
{title2=$\text{timelike current: flat; null current: vacuum pp-wave}$}
A parallel commuting <Dirac current> is causal. If timelike, the geometry locally splits as a time line times a three-dimensional Ricci-flat metric; three-dimensional Ricci flatness implies flatness. If null, the stabilizer of a parallel spinor in $\operatorname{Spin}(1,3)$ contains only null rotations, and the local vacuum metric is $ds^2=-2du\,dv+dx^2+dy^2+H(u,x,y)du^2$ with $H_{xx}+H_{yy}=0$. Constant spinors obeying $\gamma^+\epsilon=0$ are parallel. These are local statements; global quotients must preserve a compatible <spin structure>.
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