= Solution
First, the inverse <metric tensor> printed in the PDF has a missing factor: its lower-left entry must be $g^{i0}=\alpha^{-2}\beta^i$, symmetric with $g^{0i}$. This correction follows either from symmetry or direct block multiplication and does not change the evolution equation used here.
Let $D_t=\partial_t-\beta^k\partial_k$. Contract the spatial <metric tensor> evolution equation with $\gamma^{ij}$. Both shift-gradient terms contribute $\partial_k\beta^k$, giving
$$
\gamma^{ij}D_t\gamma_{ij}-2\partial_k\beta^k=-2\alpha\gamma^{ij}K_{ij}=-2\alpha K.
$$
The <Jacobi determinant derivative formula> applies to every derivative of $\gamma=\det(\gamma_{ij})$ and gives $D_t\gamma=\gamma\gamma^{ij}D_t\gamma_{ij}$. Therefore \b[the spatial determinant evolves as]
$$
\boxed{\partial_t\gamma-\beta^k\partial_k\gamma-2\gamma\partial_k\beta^k=-2\alpha\gamma K.}
$$
Equivalently, the <spatial volume evolution identity> is $D_t\sqrt\gamma=\sqrt\gamma(\partial_k\beta^k-\alpha K)$, with $K$ the trace of the <extrinsic curvature of a spatial hypersurface>.
Back to article page
