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

Proposition: Euler derivative of the cubic density

Explicit cubic density and source audit › First action-to-source calculation · lines 1185–1230 · proof included or linked

Em01-F@3#prop:cubicEuler, Em01-F@3#eq:cubicvariation, Em01-F@3#eq:J2evaluated

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For compact $h, k$ ,

δ \int ℓ_{3} [h] d^{4} x = \int J_{2} [h]_{μν} k^{μν} d^{4} x,

where

J_{2} [h]_{μν} = 2 [V_{2} (h)_{μν} + b_{1} [h] V_{1} (h)_{μν} - h_{μ}^{α} V_{1} (h)_{α ν} - h_{ν}^{α} V_{1} (h)_{μα}] .

Proof. Expand (eq:EHvariation) in the field variable $g = η + λh$ . Its lowered Euler output is

\frac{2}{λ} η_{μ ρ} η_{ν σ} - det g g^{ρ α} g^{σ β} G_{α β} [g] .

The coefficient of $λ^{2}$ inside the last density is

V_{2} + b_{1} V_{1} - h_{μ}^{α} V_{1, α ν} - h_{ν}^{α} V_{1, μα} .

Multiplication by $2/ λ$ gives (eq:J2evaluated) at order $λ$ . The coefficient of the action at that order is precisely (eq:ell3). Formal coefficient extraction commutes with compact variation, proving (eq:cubicvariation). The quadratic gauge-fixing term has no contribution at this order.

Displayed formulas in this passage

Em01-F@3#eq:cubicvariation · line 1188
$δ \int ℓ_{3} [h] d^{4} x = \int J_{2} [h]_{μν} k^{μν} d^{4} x,$
Em01-F@3#eq:J2evaluated · line 1195
$J_{2} [h]_{μν} = 2 [V_{2} (h)_{μν} + b_{1} [h] V_{1} (h)_{μν} - h_{μ}^{α} V_{1} (h)_{α ν} - h_{ν}^{α} V_{1} (h)_{μα}] .$
unlabeled · line 1212
$\frac{2}{λ} η_{μ ρ} η_{ν σ} - det g g^{ρ α} g^{σ β} G_{α β} [g] .$
unlabeled · line 1220
$V_{2} + b_{1} V_{1} - h_{μ}^{α} V_{1, α ν} - h_{ν}^{α} V_{1, μα} .$

Proposition: Euler derivative of the cubic density

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