= Harmonic lapse evolution
{title2=$(\partial_t-\beta^i\partial_i)\alpha=-\alpha^2K$}
For a positive <lapse function> $\alpha$, <shift vector> $\beta^i$, and the convention $K_{ij}=-(\partial_t\gamma_{ij}-\mathcal L_\beta\gamma_{ij})/(2\alpha)$ for <extrinsic curvature of a spatial hypersurface>, the time-coordinate <harmonic slicing> condition yields the displayed advective lapse equation. Combining $\partial_t(\sqrt\gamma/\alpha)=\partial_i(\sqrt\gamma\beta^i/\alpha)$ with the <spatial volume evolution identity> proves it without setting the <shift vector> to zero.
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