 Solution

Let the symmetric metric perturbation $h^{\alpha \beta }$ be varied with compact support. Integrating the mixed derivative product by parts makes its integral equal to that of $A_{\sigma }{A}^{\sigma }$. Thus the supplied massless Fierz-Pauli action has, up to a boundary term, densityFor example, the equivalence of the mixed term follows from commuting flat-space partial derivatives after moving one derivative off $h^{\rho \sigma }$. Here $\delta h={\eta }_{\alpha \beta }\delta h^{\alpha \beta }$ and $\delta A^{\nu }={\partial }_{\mu }\delta h^{\mu \nu }$.

Integrating each variation by parts, the four terms contribute respectivelyThe symmetrization in the second term is required because the varied field is symmetric. Comparing this coefficient with the Einstein tensor perturbation above givesThus arbitrary compactly supported variations give $\delta G_{\alpha \beta }=0$, exactly the vacuum Linearized Einstein equations. The minus sign and overall normalization of the action do not change those equations; boundary conditions justify the discarded total derivatives.