Sen spinor connection (${\nabla }_i=D_i^{(3)}-\frac{1}{2}{k}_{ij}{\gamma }^j{\gamma }^{\hat{0}}$)

Restrict the spacetime spin connection to a spacelike hypersurface. With future unit normal $n$ and convention $k_{ij}=-g({\nabla }_{i}n,e_j)$, it differs from the intrinsic spin connection by the displayed extrinsic curvature term. Its spatial Dirac operator is the operator in the Witten spinor equation; replacing it by the intrinsic operator would omit the ADM momentum contribution.