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Let $a_{\mu }=n^{\nu }{\nabla }_{\nu }{n}_{\mu }$ be the normal acceleration. It is tangential by normalization. Resolving the derivative's first index into normal and tangential parts givesThe extrinsic curvature is symmetric because the normal is hypersurface orthogonal: locally $n_{\mu }=-N{\partial }_{\mu }t$, and the antisymmetric derivative of $n$ vanishes after projecting both indices tangentially.

Apply the supplied Ricci commutator to $n^{\nu }$ and project its free index. With the positive-expansion convention it givesTo identify the first two terms, expand the fully projected divergence:Tangency gives $n_{\lambda }{K}^{\lambda }{}_{\mu }=0$. Differentiating it shows that the last term is $-P^{\mu }{}_{\alpha }{K}^{\lambda }{}_{\mu }{n}^{\rho }{\nabla }_{\rho }{n}_{\lambda }=-K_{\alpha \lambda }{a}^{\lambda }$. Substituting into the commutator proves the contracted Codazzi equation with positive expansion:No assumption of geodesic normals was needed; the acceleration terms cancel rather than being discarded.

The metric-trace term in the Einstein field equations drops out of a tangential-normal projection, since $P^{\mu }{}_{\alpha }{g}_{\mu \nu }{n}^{\nu }=0$. Define the normal-frame momentum density $j_{\alpha }=-P^{\mu }{}_{\alpha }{T}_{\mu \nu }{n}^{\nu }$. The equation becomesThus Codazzi supplies the momentum constraint on each spatial slice. In flat homogeneous FRW, $K^i{}_j=H{\delta }^i{}_j$ and $H$ has no spatial gradient, so its left side vanishes. A comoving homogeneous perfect fluid likewise has $j_i=0$. The background satisfies the constraint automatically; the Friedmann expansion law is instead obtained from the normal-normal constraint.

Suitable scalar variables are lapse and shift potentials $A,B$, spatial curvature and scalar shear potentials $\psi ,E$, and a matter momentum potential $J$ with $j_i={\partial }_{i}J$. For example, chooseA density contrast and pressure perturbation can be added for the remaining scalar matter equations. With this positive-shift convention, the linear extrinsic-curvature perturbation isIts scalar shear cancels out of the momentum-constraint combination, leavingThese are the scalar momentum constraint with positive extrinsic curvature variables; a specified gauge can then remove two of the four scalar metric potentials.