 Massless Fierz-Pauli action

ID: massless-fierz-pauli-action

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

L = - \frac{1}{4} \partial_{μ} h^{ρ σ} \partial^{μ} h_{ρ σ} + \frac{1}{2} A_{μ} A^{μ} - \frac{1}{2} A^{μ} \partial_{μ} h + \frac{1}{4} \partial_{μ} h \partial^{μ} h, A_{μ} = \partial_{ν} h^{ν}_{μ} .

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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