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Linearized Ricci tensor and scalar

To first order around flat spacetime,

R_{μν} = \frac{1}{2} (\partial_{ρ} \partial_{μ} h^{ρ}_{ν} + \partial_{ρ} \partial_{ν} h^{ρ}_{μ} - □ h_{μν} - \partial_{μ} \partial_{ν} h),

and

R = \partial_{μ} \partial_{ν} h^{μν} - □ h, h = η^{μν} h_{μν} .

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