Manuscript XI-A@1 · XI-A.tex · proposition · b8211ecc

Proposition: Quadratic gravitational coefficient

One action, one gauge-fixing term, and two sign checks › Explicit quadratic action · lines 486–527 · proof included or linked

XI-A@1#prop:FP, XI-A@1#eq:FP, XI-A@1#eq:Glin

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Modulo a total divergence, the quadratic bulk coefficient of (eq:EH) is

S_{FP} [γ] = \frac{1}{2} \int_{M_{T}} (\partial_{λ} γ_{μν} \partial^{λ} γ^{μν} - 2 (\partial_{μ} γ^{μν}) (\partial^{ρ} γ_{ρ ν}) + 2 (\partial_{μ} γ^{μν}) \partial_{ν} τ - \partial_{λ} τ \partial^{λ} τ) d^{4} x .

Its Euler derivative is $2 G_{lin}^{μν} (γ)$ , where

G_{lin}^{μν} (γ) = - \frac{1}{2} □ γ ˉ^{μν} + \partial^{(μ} D^{ν)} - \frac{1}{2} η^{μν} \partial_{ρ} D^{ρ} .

Proof. Expanding (eq:ricci) to first order gives (eq:Glin). Variation with respect to the covariant metric gives

δ S_{EH} = \frac{2}{κ ^{2}} \int - q G^{μν} (q) δ q_{μν} d^{4} x .

Consequently the quadratic coefficient has derivative $2 G_{lin} (γ)$ . Integration by parts in (eq:FP) gives that same derivative. Both quadratic functionals vanish at $γ = 0$ ; integrating their equal directional derivatives along $s γ$ , $0 \leq s \leq 1$ , proves equality as bulk variational functionals. The linear Einstein–Hilbert coefficient is a divergence and has no bulk Euler derivative.

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XI-A@1#eq:FP · line 490
$S_{FP} [γ] = \frac{1}{2} \int_{M_{T}} (\partial_{λ} γ_{μν} \partial^{λ} γ^{μν} - 2 (\partial_{μ} γ^{μν}) (\partial^{ρ} γ_{ρ ν}) + 2 (\partial_{μ} γ^{μν}) \partial_{ν} τ - \partial_{λ} τ \partial^{λ} τ) d^{4} x .$
XI-A@1#eq:Glin · line 503
$G_{lin}^{μν} (γ) = - \frac{1}{2} □ γ ˉ^{μν} + \partial^{(μ} D^{ν)} - \frac{1}{2} η^{μν} \partial_{ρ} D^{ρ} .$
unlabeled · line 515
$δ S_{EH} = \frac{2}{κ ^{2}} \int - q G^{μν} (q) δ q_{μν} d^{4} x .$

Proposition: Quadratic gravitational coefficient

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