= Metric determinant in a 3+1 decomposition
{title2=$\det g=-\alpha^2\det\gamma$}
For a positive <lapse function> $\alpha$, <shift vector> $\beta^i$ and spatial <induced metric> $\gamma_{ij}$, the block metric has $g_{00}=-\alpha^2+\gamma_{ij}\beta^i\beta^j$ and $g_{0i}=\gamma_{ij}\beta^j$. Its block determinant is $g=-\alpha^2\det\gamma$. Therefore the <metric volume tensor> contracted with the future <unit normal> restricts to the spatial volume tensor: $n^\mu\epsilon_{\mu ijk}=\sqrt{\det\gamma}\,[ijk]$.
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