Manuscript VII-B@1 · VII-B.tex · section · be632835

Fixed gravitational density representative

An explicit action and vertex ledger › Fixed gravitational density representative · lines 1417–1469

VII-B@1#eq:grav-density, VII-B@1#eq:FP

For $g = η + κh$ , let

Γ^{ρ}_{μν} (g) = \frac{1}{2} g^{ρ λ} (\partial_{μ} g_{λ ν} + \partial_{ν} g_{λ μ} - \partial_{λ} g_{μν}) .

Choose the first-derivative Einstein density representative

L_{grav} (κ) = - \frac{2}{κ ^{2}} A^{μν} (κ) (Γ^{ρ}_{μ σ} Γ^{σ}_{ν ρ} - Γ^{ρ}_{μν} Γ^{σ}_{ρ σ}) .

The curvature-sign convention is the one for which this is the Einstein density modulo a total divergence. The representative (eq:grav-density), rather than its equivalence class modulo divergences, is used before switching.

Because $Γ = O (κ)$ , the density has a regular formal expansion

L_{grav} (κ) = L_{grav, 0} + κ L_{grav, 1} + κ^{2} L_{grav, 2} + O (κ^{3}) .

Its quadratic part, modulo an unswitched divergence, is

L_{grav, 0} = \frac{1}{2} \partial_{λ} h_{μν} \partial^{λ} h^{μν} - (\partial_{μ} h^{μν}) (\partial^{ρ} h_{ρ ν}) + (\partial_{μ} h^{μν}) \partial_{ν} h - \frac{1}{2} \partial_{λ} h \partial^{λ} h .

Displayed formulas in this passage

VII-B@1#eq:grav-density · line 1431
$L_{grav} (κ) = - \frac{2}{κ ^{2}} A^{μν} (κ) (Γ^{ρ}_{μ σ} Γ^{σ}_{ν ρ} - Γ^{ρ}_{μν} Γ^{σ}_{ρ σ}) .$
VII-B@1#eq:FP · line 1458
$L_{grav, 0} = \frac{1}{2} \partial_{λ} h_{μν} \partial^{λ} h^{μν} - (\partial_{μ} h^{μν}) (\partial^{ρ} h_{ρ ν}) + (\partial_{μ} h^{μν}) \partial_{ν} h - \frac{1}{2} \partial_{λ} h \partial^{λ} h .$
unlabeled · line 1420
$Γ^{ρ}_{μν} (g) = \frac{1}{2} g^{ρ λ} (\partial_{μ} g_{λ ν} + \partial_{ν} g_{λ μ} - \partial_{λ} g_{μν}) .$
unlabeled · line 1449
$L_{grav} (κ) = L_{grav, 0} + κ L_{grav, 1} + κ^{2} L_{grav, 2} + O (κ^{3}) .$