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## Main Question or Discussion Point

Hi i have been following Hobson in their attempt to derive the einstein tensor, I have split the varied action into three terms and want to factor out [itex] \delta(g^{\mu\nu})[/itex] terms.

The Riemann tensor [itex] R_{\mu \nu}[/itex] must be expanded to [itex]R^{\rho}_{\mu \nu p}[/itex] and then contracted back to the original form. To do this should one simply multiply by [itex](g^{\alpha\sigma}g_{\alpha \rho})(g_{\alpha\sigma}g^{\alpha\rho} )[/itex] and expand with the first bracket and then contract down with the second?

The Riemann tensor [itex] R_{\mu \nu}[/itex] must be expanded to [itex]R^{\rho}_{\mu \nu p}[/itex] and then contracted back to the original form. To do this should one simply multiply by [itex](g^{\alpha\sigma}g_{\alpha \rho})(g_{\alpha\sigma}g^{\alpha\rho} )[/itex] and expand with the first bracket and then contract down with the second?