Manuscript XII-C@1 · XII-C.tex · theorem · dead1004

Theorem: Actual density-weighted cubic source

C1: the density representative and actual cubic source › All first and second ramp derivatives · lines 718–760 · proof included or linked

XII-C@1#thm:actualsource, XII-C@1#eq:actualsource, XII-C@1#eq:time-layer

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The selected cubic vertex

V_{3, D} [γ] = \int_{M_{T}} σ_{ε} ℓ_{3} [γ] d^{4} x

has Euler source

J_{3, D}^{μν} = σ_{ε} J_{3}^{μν} + D_{σ}_{ε}^{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} .

For the time-only ramp,

D_{σ}_{ε}^{μν} = 2 C_{2}^{μν 00} σ_{ε}^{''} - 2 M_{2}^{μν 0} σ_{ε}^{'} .

These are all ramp-derivative terms in the cubic source.

Proof. The extra term in (eq:weighted-full-variation) is

\frac{2}{κ} C^{μν α β} (q) (\partial_{α} \partial_{β} σ_{ε} - Γ^{ρ}_{α β} (q) \partial_{ρ} σ_{ε}) .

Its coefficient of $κ$ is (eq:actualsource), using (eq:Ccoefficients) and (eq:M2). The Einstein term at that coefficient is $σ_{ε} J_{3}$ . No derivative of the fixed weight beyond order two occurs.

Displayed formulas in this passage

XII-C@1#eq:actualsource · line 726
$J_{3, D}^{μν} = σ_{ε} J_{3}^{μν} + D_{σ}_{ε}^{μν}, D_{σ}_{ε}^{μν} = 2 C_{2}^{μν α β} \partial_{α} \partial_{β} σ_{ε} - 2 M_{2}^{μν ρ} \partial_{ρ} σ_{ε} .$
XII-C@1#eq:time-layer · line 737
$D_{σ}_{ε}^{μν} = 2 C_{2}^{μν 00} σ_{ε}^{''} - 2 M_{2}^{μν 0} σ_{ε}^{'} .$
unlabeled · line 721
$V_{3, D} [γ] = \int_{M_{T}} σ_{ε} ℓ_{3} [γ] d^{4} x$
unlabeled · line 748
$\frac{2}{κ} C^{μν α β} (q) (\partial_{α} \partial_{β} σ_{ε} - Γ^{ρ}_{α β} (q) \partial_{ρ} σ_{ε}) .$

Theorem: Actual density-weighted cubic source

Displayed formulas in this passage

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