Witten spinor equation (${\gamma }^i{\nabla }_iϵ=0$)

This elliptic equation on a spacelike spin hypersurface uses the restriction of the spacetime spin connection, including extrinsic curvature. The spinor tends to a prescribed constant at spatial infinity. It removes the negative derivative square in the divergence of the Nester two-form. It is weaker than the spacetime Killing spinor equation; generic positive-energy initial data need not admit any Killing spinor.

On complete smooth asymptotically flat spin initial data without inner boundary, satisfying the Einstein constraints and the dominant energy condition, the Witten spinor equation admits a spinor approaching each prescribed asymptotic constant, under the standard weighted decay assumptions. The integrated Witten-Nester identity gives a coercive estimate and excludes a decaying homogeneous kernel. Weighted elliptic solvability then supplies the desired correction to a spinor extended from infinity. Positivity of the boundary integral is a conditional identity until this analytic existence step is supplied; see the rigorous analytic proof.

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.