= Witten-Nester identity
{c}
{title2=$D_ib^i=2(|\nabla_i\epsilon|^2-|\gamma^i\nabla_i\epsilon|^2)+G_{\mu\nu}n^\mu K^\nu$}
In mostly-plus signature orient the commuting <Dirac current> to the future by $\bar\epsilon=-\epsilon^\dagger\gamma^{\hat0}$. 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.
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