= Spatial volume evolution identity
{title2=$(\partial_t-\beta^i\partial_i)\sqrt\gamma=\sqrt\gamma(\partial_i\beta^i-\alpha K)$}
In a <3+1 decomposition of spacetime> with $K_{ij}=-(\partial_t\gamma_{ij}-\mathcal L_\beta\gamma_{ij})/(2\alpha)$, contract the metric evolution with $\gamma^{ij}$ and use the <Jacobi determinant derivative formula>. Here $\gamma=\det\gamma_{ij}$, $K=\gamma^{ij}K_{ij}$, $\alpha$ is the <lapse function>, and $\beta$ is the <shift vector>. The identity separates spatial transport and coordinate compression from the geometric volume change caused by <extrinsic curvature>.
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