= Static zero-energy parallel-spinor rigidity
{title2=$E=0\Rightarrow\nabla_\mu\epsilon=0$}
The vacuum <Witten-Nester identity> makes a zero-energy Witten spinor parallel under the <Sen spinor connection> on a complete asymptotically flat slice. In an orthogonal static slicing, extrinsic curvature vanishes, so this spinor is intrinsically parallel. Contracting its spin-curvature integrability condition gives zero spatial <Ricci tensor>; in three dimensions this implies flatness. The static vacuum equation $D_iD_jN=N R_{ij}(h)$ then gives a parallel lapse gradient, which asymptotic normalization forces to vanish. The time-independent spinor consequently extends to a spacetime <parallel spinor>, with <Minkowski spacetime> the complete regular asymptotically flat example.
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