Editing Einstein–Hilbert action (section)

===Variation of the determinant===
[[Jacobi's formula]], the rule for differentiating a [[determinant#Derivative|determinant]], gives:

:<math>\delta g = \delta \det(g_{\mu\nu}) = g g^{\mu\nu} \delta g_{\mu\nu}</math>,

or one could transform to a coordinate system where <math>g_{\mu\nu}</math> is diagonal and then apply the product rule to differentiate the product of factors on the [[main diagonal]]. Using this we get

:<math>\delta \sqrt{-g} = -\frac{1}{2\sqrt{-g}}\delta g = \frac{1}{2} \sqrt{-g} \left( g^{\mu\nu} \delta g_{\mu\nu} \right) = -\frac{1}{2} \sqrt{-g} \left( g_{\mu\nu} \delta g^{\mu\nu} \right)</math>

In the last equality we used the fact that

:<math>g_{\mu\nu}\delta g^{\mu\nu} = -g^{\mu\nu} \delta g_{\mu\nu}</math>

which follows from the rule for differentiating the inverse of a matrix

:<math>\delta g^{\mu\nu} = - g^{\mu\alpha} \left( \delta g_{\alpha\beta} \right) g^{\beta\nu}</math>.

Thus we conclude that

{{NumBlk|:|<math>\frac{1}{\sqrt{-g}} \frac{\delta \sqrt{-g}}{\delta g^{\mu\nu} } = -\frac{1}{2} g_{\mu\nu}</math>.|{{EquationRef|4}}}}