Manuscript Em01-F@3 · Em01-F-v3.tex · section · 225e59af

First action-to-source calculation

Explicit cubic density and source audit › First action-to-source calculation · lines 1230–1305

Em01-F@3#eq:firstjetdensity, Em01-F@3#eq:firstjetvariation

For an independent density-level check, define

H_{μν} (A, B) = A^{ρ}_{μ σ} B^{σ}_{ν ρ} - A^{ρ}_{μν} B^{σ}_{ρ σ} .

The coordinate Ricci formula and metric compatibility give

- \frac{2}{λ ^{2}} - det g R [g] = - \frac{2}{λ ^{2}} a_{λ}^{μν} H_{μν} (Γ, Γ) + \partial_{ρ} V^{ρ},

where

V^{ρ} = - \frac{2}{λ ^{2}} a_{λ}^{μν} Γ^{ρ}_{μν} + \frac{2}{λ ^{2}} a_{λ}^{ρ μ} Γ^{ν}_{μν} .

Hence $ℓ_{3}$ , modulo a displayed divergence, equals the first-jet cubic density

ℓ_{3}^{(1)} = - 2 [a_{1}^{μν} [h] H_{μν} (C, C) + η^{μν} (H_{μν} (C, C_{2}) + H_{μν} (C_{2}, C))] .

Here the superscript $(1)$ refers to first-jet dependence, not coupling order.

For a variation $k$ ,

δ C = C (k), δ C_{2} = - \frac{1}{2} (k^{ρ σ} S (h)_{σ μν} + h^{ρ σ} S (k)_{σ μν}) = D (h, k)_{μν}^{ρ} .

Thus its entire directional variation is

δ ℓ_{3}^{(1)} = - 2 {a_{1}^{μν} [k] H_{μν} (C, C) + a_{1}^{μν} [h] [H_{μν} (C (k), C) + H_{μν} (C, C (k))] + η^{μν} [H_{μν} (C (k), C_{2}) + H_{μν} (C, D (h, k)) + H_{μν} (D (h, k), C) + H_{μν} (C_{2}, C (k))]} .

The exact divergence identity proves that its compactly integrated variation is also (eq:cubicvariation). This supplies a second density representation for checking the source. It is valid here because the weight is constant. Multiplication by a nonconstant gravitational weight would generate additional terms from the divergence and is not authorized by this check.

Displayed formulas in this passage

Em01-F@3#eq:firstjetdensity · line 1256
$ℓ_{3}^{(1)} = - 2 [a_{1}^{μν} [h] H_{μν} (C, C) + η^{μν} (H_{μν} (C, C_{2}) + H_{μν} (C_{2}, C))] .$
Em01-F@3#eq:firstjetvariation · line 1282
$δ ℓ_{3}^{(1)} = - 2 {a_{1}^{μν} [k] H_{μν} (C, C) + a_{1}^{μν} [h] [H_{μν} (C (k), C) + H_{μν} (C, C (k))] + η^{μν} [H_{μν} (C (k), C_{2}) + H_{μν} (C, D (h, k)) + H_{μν} (D (h, k), C) + H_{μν} (C_{2}, C (k))]} .$
unlabeled · line 1233
$H_{μν} (A, B) = A^{ρ}_{μ σ} B^{σ}_{ν ρ} - A^{ρ}_{μν} B^{σ}_{ρ σ} .$
unlabeled · line 1240
$- \frac{2}{λ ^{2}} - det g R [g] = - \frac{2}{λ ^{2}} a_{λ}^{μν} H_{μν} (Γ, Γ) + \partial_{ρ} V^{ρ},$
unlabeled · line 1248
$V^{ρ} = - \frac{2}{λ ^{2}} a_{λ}^{μν} Γ^{ρ}_{μν} + \frac{2}{λ ^{2}} a_{λ}^{ρ μ} Γ^{ν}_{μν} .$
unlabeled · line 1271
$δ C = C (k), δ C_{2} = - \frac{1}{2} (k^{ρ σ} S (h)_{σ μν} + h^{ρ σ} S (k)_{σ μν}) = D (h, k)_{μν}^{ρ} .$

First action-to-source calculation

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