= Trace of extrinsic curvature with a negative shift
{title2=$K=-\frac1{2N}(\partial_t\log\gamma+2D_iN^i)$}
For the threading convention $dx^i-N^i dt$ and $K_{ij}=-(\dot\gamma_{ij}+D_iN_j+D_jN_i)/(2N)$, contract with $\gamma^{ij}$. <Metric compatibility> gives $\gamma^{ij}D_iN_j=D_iN^i$, and the <Jacobi determinant derivative formula> gives $\gamma^{ij}\dot\gamma_{ij}=\partial_t\log\det\gamma$. With zero shift, $-K/3=(3N)^{-1}\partial_t\log\sqrt\gamma$ is the <local volume Hubble parameter>. Reversing the shift convention reverses both the normal's spatial component and the shift terms; the two conventions must not be mixed.
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