Manuscript XII-D@1 · XII-D.tex · lemma · 308a2080

Lemma: Finite source variation

D5: the exact signed plateau variation › The finite cubic identity and its sign · lines 1098–1129 · proof included or linked

XII-D@1#lem:finite-source, XII-D@1#eq:finite-source

For every smooth $γ$ and every $ξ \in g_{U}$ ,

J_{D} [γ + K ξ] - J_{D} [γ] = L_{ξ}^{dens} (A γ) - A s_{ξ} [γ] .

Proof. Equation (eq:layer-variation) reduces the assertion to the unweighted local coefficient identity. For the flow generated by $κ ξ$ ,

\frac{F ^{*} q - η}{κ} = γ + K ξ + κ s_{ξ} [γ] + O (κ^{2}) .

Naturality of the Einstein tensor density gives the same pullback for its Euler derivative. Comparing the coefficient of $κ$ in

e_{EH} = A γ + κ J_{3} [γ] + O (κ^{2})

gives

A s_{ξ} + J_{3} [γ + K ξ] = J_{3} [γ] + L_{ξ}^{dens} (A γ) .

This proves (eq:finite-source).

Displayed formulas in this passage

XII-D@1#eq:finite-source · line 1101
$J_{D} [γ + K ξ] - J_{D} [γ] = L_{ξ}^{dens} (A γ) - A s_{ξ} [γ] .$
unlabeled · line 1113
$\frac{F ^{*} q - η}{κ} = γ + K ξ + κ s_{ξ} [γ] + O (κ^{2}) .$
unlabeled · line 1120
$e_{EH} = A γ + κ J_{3} [γ] + O (κ^{2})$
unlabeled · line 1124
$A s_{ξ} + J_{3} [γ + K ξ] = J_{3} [γ] + L_{ξ}^{dens} (A γ) .$