Manuscript XIV-G@1 · XIV-G.tex · input · 791c6644

Input statement: Density representative and actual source

G0: copied inputs and the unchanged stop › Earlier analytic handoffs · lines 712–754

XIV-G@1#input:XIIC, XIV-G@1#eq:source

For $q (ρ) = η + ρ γ$ ,

ℓ_{3} [γ] = - 2 [ρ^{3}] (- det q (ρ) R [q (ρ)])

is the fixed unintegrated cubic representative, with

R_{μν} = \partial_{ρ} Γ^{ρ}_{μν} - \partial_{ν} Γ^{ρ}_{μ ρ} + Γ^{ρ}_{ρ λ} Γ^{λ}_{μν} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ ρ} .

Its actual weighted source is

J_{D}^{μν} D_{σ}_{ε}^{μν} = EL_{γ} (σ_{ε} ℓ_{3})^{μν} = σ_{ε} J_{3}^{μν} + D_{σ}_{ε}^{μν}, = 2 C_{2}^{μν α β} \partial_{α} \partial_{β} σ_{ε} - 2 M_{2}^{μν ρ} \partial_{ρ} σ_{ε} .

Here $C_{2}$ is the quadratic coefficient of

- q (q^{μν} q^{α β} - q^{μ (α} q^{β) ν}),

and

M_{2}^{μν ρ} = C_{0}^{μν α β} Γ_{2}^{ρ}_{α β} + C_{1}^{μν α β} Γ_{1}^{ρ}_{α β} .

The source is density weighted before variation, at cubic order only XIIC.

Displayed formulas in this passage

XIV-G@1#eq:source · line 728
$J_{D}^{μν} D_{σ}_{ε}^{μν} = EL_{γ} (σ_{ε} ℓ_{3})^{μν} = σ_{ε} J_{3}^{μν} + D_{σ}_{ε}^{μν}, = 2 C_{2}^{μν α β} \partial_{α} \partial_{β} σ_{ε} - 2 M_{2}^{μν ρ} \partial_{ρ} σ_{ε} .$
unlabeled · line 715
$ℓ_{3} [γ] = - 2 [ρ^{3}] (- det q (ρ) R [q (ρ)])$
unlabeled · line 720
$R_{μν} = \partial_{ρ} Γ^{ρ}_{μν} - \partial_{ν} Γ^{ρ}_{μ ρ} + Γ^{ρ}_{ρ λ} Γ^{λ}_{μν} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ ρ} .$
unlabeled · line 741
$- q (q^{μν} q^{α β} - q^{μ (α} q^{β) ν}),$
unlabeled · line 747
$M_{2}^{μν ρ} = C_{0}^{μν α β} Γ_{2}^{ρ}_{α β} + C_{1}^{μν α β} Γ_{1}^{ρ}_{α β} .$