Hi I am trying to derive the E field equation and am stuck using the Jacobi formula, is this correct: $$\delta \det g_{\mu \nu} = Tr(ADJ(g_{\mu\nu}\delta g_{\mu\nu})=\det(g_{\mu\nu})Tr(g^{\mu\nu}\delta g_{\mu \nu}) $$ Then how can we remove the trace, or should it be: $$\delta \det g_{\mu \nu} =\det(g_{\mu\nu})Tr(g^{-1}\delta g)=\det(g_{\mu\nu})(g\delta g)^{\mu}_{\mu}=\det(g_{\mu\nu})(g\delta g)^{\mu}_{\mu}=\det(g_{\mu\nu})(g^{\mu\nu}\delta g_{\mu\nu}) $$
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You should be careful not to mix symbolic and index notation. $ \text{Tr}(g^{\mu \nu}\delta g_{\mu \nu}) $ does not make sense since $g^{\mu \nu}\delta g_{\mu \nu}$ is just a number. The correct symbolic notation would be: $$\delta \text{det}(\mathbf{g})=\text{Tr}(\text{adj}(\mathbf{g})\delta \mathbf{g})=\text{det}(\mathbf{g})\text{Tr}(\mathbf{g}^{-T}\delta \mathbf{g}))$$ since $\text{adj}(\mathbf{g})=\text{det}(\mathbf{g})\mathbf{g}^{-T}$. Now you can transform to index notation and Einstein convention: $$\delta \text{det}(g_{\mu \nu})=\text{det}(g_{\mu \nu})(g^{\mu \nu} \delta g_{\mu \nu}) $$ |
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