= Massless Fierz-Pauli action
{title2=$\delta S=-\int d^4x\,G^{(1)}_{\alpha\beta}\delta h^{\alpha\beta}$}
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, is
$$
\mathcal L=-\frac14\partial_\mu h^{\rho\sigma}\partial^\mu h_{\rho\sigma}
+\frac12A_\mu A^\mu-\frac12A^\mu\partial_\mu h
+\frac14\partial_\mu h\partial^\mu h,\qquad A_\mu=\partial_\nu h^\nu{}_\mu.
$$
Varying 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.
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