Manuscript Em01-D@2 · Em01-D-v2.tex · section · d3b49151

The unintegrated cubic source

Finite source and density specifications › The unintegrated cubic source · lines 500–569

Em01-D@2#eq:source, Em01-D@2#eq:sourceconversion

For a formal metric $q_{s} = η + s γ$ , fix

Γ^{ρ}_{μν} (q) = \frac{1}{2} q^{ρ λ} (\partial_{μ} q_{ν λ} + \partial_{ν} q_{μ λ} - \partial_{λ} q_{μν}),

R_{μν} (q) = \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},

and $R (q) = q^{μν} R_{μν} (q)$ . Declare

ℓ_{3} = - 2 [s^{3}] (- det q_{s} R (q_{s})), J_{D} = EL_{γ} (σ ℓ_{3}) .

No integration-by-parts replacement of the density is made before this variation.

Let $X_{I, γ}$ , $∣ γ ∣ \leq 2$ , denote ordinary, not factorial-normalized, derivative jets:

\partial^{γ} γ_{μν} = I \sum (e_{I})_{μν} X_{I, γ} .

Mixed derivatives use their common multiindex.

The coordinate Euler components are

E_{I} = ∣ γ ∣ \leq 2 \sum (- \partial)^{γ} \frac{\partial ( σ ℓ _{3} )}{\partial X _{I, γ}} .

Let

G_{I J} = (e_{I})_{μν} (e_{J})^{μν} .

The propagated source components are

J^{L} = (G^{- 1})^{L I} E_{I} = (e^{L})^{μν} η_{μ ρ} η_{ν σ} J_{D}^{ρ σ} .

Thus the coordinate Euler gradient is not substituted for the metric-valued propagated source.

Displayed formulas in this passage

Em01-D@2#eq:source · line 523
$ℓ_{3} = - 2 [s^{3}] (- det q_{s} R (q_{s})), J_{D} = EL_{γ} (σ ℓ_{3}) .$
Em01-D@2#eq:sourceconversion · line 558
$J^{L} = (G^{- 1})^{L I} E_{I} = (e^{L})^{μν} η_{μ ρ} η_{ν σ} J_{D}^{ρ σ} .$
unlabeled · line 503
$Γ^{ρ}_{μν} (q) = \frac{1}{2} q^{ρ λ} (\partial_{μ} q_{ν λ} + \partial_{ν} q_{μ λ} - \partial_{λ} q_{μν}),$
unlabeled · line 513
$R_{μν} (q) = \partial_{ρ} Γ^{ρ}_{μν} - \partial_{ν} Γ^{ρ}_{μ ρ} + Γ^{ρ}_{ρ λ} Γ^{λ}_{μν} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ ρ},$
unlabeled · line 538
$\partial^{γ} γ_{μν} = I \sum (e_{I})_{μν} X_{I, γ} .$
unlabeled · line 546
$E_{I} = ∣ γ ∣ \leq 2 \sum (- \partial)^{γ} \frac{\partial ( σ ℓ _{3} )}{\partial X _{I, γ}} .$
unlabeled · line 554
$G_{I J} = (e_{I})_{μν} (e_{J})^{μν} .$