Nester two-form ($B^{\mu \nu }=\overline{ϵ}{\gamma }^{\mu \nu \rho }{\nabla }_{\rho }ϵ-\overline{{\nabla }_{\rho }ϵ}{\gamma }^{\mu \nu \rho }ϵ$)

This antisymmetric spinor bilinear converts the positive energy theorem into a boundary-charge identity. Its divergence is $2({\nabla }_{\nu }\overline{ϵ}){\gamma }^{\mu \nu \rho }{\nabla }_{\rho }ϵ+G^{\mu \nu }\overline{ϵ}{\gamma }_{\nu }ϵ$ in the mostly-plus conventions specified here. On a spacelike hypersurface the first term is twice the difference between the squared spatial covariant derivative and the squared spatial Dirac operator. The Witten spinor equation removes the negative square, while the dominant energy condition controls the curvature term through the Einstein constraints.

For asymptotically constant spinors, the leading intrinsic spin connection term gives half the ADM metric integrand times $K_{\infty }^0$. The Sen spinor connection correction gives $-(k_{ij}-k{h}_{ij})K_{\infty }^j$. The constant-spinor derivative term integrates to zero by antisymmetry of the gamma matrices, and the remaining quadratic terms vanish under the standard decay. This identifies the Nester two-form flux with the ADM energy-momentum contraction, including momentum rather than just energy.

In mostly-plus signature orient the commuting Dirac current to the future by $\overline{ϵ}=-{ϵ}^{†}{\gamma }^{\hat{0}}$. For the Nester two-form, put $b^i=n_{\mu }{B}^{\mu i}$. Its covariant divergence has a spinor-gradient term and the Einstein tensor contraction of spin curvature. Projecting onto the hypersurface gives the displayed identity with positive spinor norm. The Witten spinor equation removes the negative square, and the dominant energy condition makes the matter term nonnegative after the Einstein constraints are imposed.

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.