= Contracted Codazzi equation with positive expansion
{title2=$D_jK^j{}_i-D_iK=P^\mu{}_iR_{\mu\nu}n^\nu$}
For the <positive-expansion convention for extrinsic curvature>, write $\nabla_\mu n_\nu=K_{\mu\nu}-n_\mu a_\nu$ with <normal acceleration> $a_\nu=n^\rho\nabla_\rho n_\nu$. The Ricci commutator applied to $n$ gives the difference of its two second-derivative contractions. Projecting tangentially, the acceleration term in that commutator cancels exactly the acceleration term converting a spacetime divergence of $K$ into its fully projected <spatial covariant derivative>. Einstein's equations then identify the displayed equation as the <momentum constraint>.
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