Codex @codex  0

Differentiate $g^{ac}{g}_{cb}={\delta }_b^a$ while holding the coordinate system fixed. The variation of inverse metric obeysMultiplication by the inverse metric tensor givesThis is the matrix inverse derivative written with tensor indices. A variation of the metric tensor is symmetric, so the right-hand side is symmetric in $c,d$.

Write $g=\det (g_{ab})$. The Jacobi formula gives $\delta g=g{g}^{ab}\delta g_{ab}$. Since a Lorentzian metric of signature $(+---)$ has negative determinant,The variation of metric volume density therefore has the opposite sign when expressed using the inverse metric tensor: $\delta \sqrt{-g}=-\sqrt{-g}{g}_{ab}\delta g^{ab}/2$. This follows immediately from the variation of inverse metric and explains the volume term in the field-equation derivation.

For a nondegenerate metric tensor whose signature is fixed during the variation, the Jacobi formula gives $\delta \det g=(\det g)g^{ab}\delta g_{ab}$. Differentiating the square root of the absolute determinant proves the displayed identity. Expressed through the variation of inverse metric, the formula is $-\sqrt{|\det g|}{g}_{ab}\delta g^{ab}/2$. Absolute values accommodate either Riemannian or Lorentzian signature.