Manuscript XIV-F@1 · XIV-F.tex · input · bae5a40b

Input statement: Fixed cubic representative and actual source

F0: copied analytic inputs › XII-E and XII-C: completion and source conventions · lines 643–683

XIV-F@1#input:XIIC

For $q (ρ) = η + ρ γ$ , the fixed unintegrated cubic representative is

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

The curvature convention is

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} .

The actual source is

J_{D} = EL_{γ} (σ_{ε} ℓ_{3}) = σ_{ε} J_{3} + D_{σ}_{ε},

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

unlabeled · line 647
$ℓ_{3} [γ] = - 2 [ρ^{3}] (- det q (ρ) R [q (ρ)]) .$
unlabeled · line 652
$R_{μν} = \partial_{ρ} Γ^{ρ}_{μν} - \partial_{ν} Γ^{ρ}_{μ ρ} + Γ^{ρ}_{ρ λ} Γ^{λ}_{μν} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ ρ} .$
unlabeled · line 660
$J_{D} = EL_{γ} (σ_{ε} ℓ_{3}) = σ_{ε} J_{3} + D_{σ}_{ε},$
unlabeled · line 664
$D_{σ}_{ε}^{μν} = 2 C_{2}^{μν α β} \partial_{α} \partial_{β} σ_{ε} - 2 M_{2}^{μν ρ} \partial_{ρ} σ_{ε} .$
unlabeled · line 670
$- q (q^{μν} q^{α β} - q^{μ (α} q^{β) ν}),$
unlabeled · line 676
$M_{2}^{μν ρ} = C_{0}^{μν α β} Γ_{2}^{ρ}_{α β} + C_{1}^{μν α β} Γ_{1}^{ρ}_{α β} .$