Massless Fierz-Pauli action ($\delta S=-\int d^{4}x{G}_{\alpha \beta }^{(1)}\delta h^{\alpha \beta }$)

A massless Fierz-Pauli action is the quadratic flat-background action for a symmetric metric perturbation that yields the vacuum Linearized Einstein equations. One normalization, up to a total derivative, isVarying the symmetric field with compact support gives the displayed linearized Einstein tensor coefficient. Its linearized coordinate-gauge symmetry is consistent with the contracted Bianchi identity. It is the massless gravitational action, distinct from imposing the massive Fierz-Pauli equations with five polarizations.