Inverse metric volume variation
= Inverse metric volume variation
{title2=$\delta\sqrt{|g|}=-\tfrac12\sqrt{|g|}g_{ab}\delta g^{ab}$}
For a nondegenerate <metric tensor>, differentiating $g_{ac}g^{cb}=\delta_a{}^b$ gives $\delta g_{ab}=-g_{ac}g_{bd}\delta g^{cd}$. Combining this with the <Jacobi determinant derivative formula> proves the displayed volume-density variation. It is the determinant contribution when defining a <stress-energy tensor> by varying the inverse <metric tensor>.