Manuscript XII-B@1 · XII-B.tex · lemma · 229bc869

Lemma: Pure-trace cubic density

An explicit Einstein–Hilbert layer discrepancy › The cubic density on the pure-trace sector · lines 631–681 · proof included or linked

XII-B@1#lem:tracedensity, XII-B@1#eq:tracedensity, XII-B@1#eq:traceEuler

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For the displayed Einstein–Hilbert density, its coefficient of $κ^{1}$ on this sector is

ℓ_{3} [ψ η] = 3 ψ \partial_{μ} ψ \partial^{μ} ψ .

Its Euler derivative is

EL_{ψ} (ℓ_{3} [ψ η]) = - 3 \partial_{μ} ψ \partial^{μ} ψ - 6 ψ □ ψ = η_{μν} J_{3}^{μν} [ψ η] .

Proof. Computing the Christoffel symbols of $e^{2 w} η$ gives

- q R [q] = - 6 e^{2 w} (□ w + \partial_{μ} w \partial^{μ} w) .

With $e^{2 w} = 1 + κ ψ$ , this is

- 3 κ □ ψ + \frac{3 κ ^{2}}{2 ( 1 + κ ψ )} \partial_{μ} ψ \partial^{μ} ψ .

Therefore the signed density is exactly

- \frac{2}{κ ^{2}} - q R [q] = \frac{6}{κ} □ ψ - \frac{3}{1 + κ ψ} \partial_{μ} ψ \partial^{μ} ψ .

Its coefficient of $κ$ is (eq:tracedensity).

Direct variation gives

3 (\partial ψ)^{2} - \partial_{μ} (6 ψ \partial^{μ} ψ) = - 3 (\partial ψ)^{2} - 6 ψ □ ψ .

Variation along $δ γ_{μν} = η_{μν} δ ψ$ contracts the full cubic Euler source with $η_{μν}$ , proving the last equality. It agrees with the pure-trace coefficient in XI-G XIG.

Displayed formulas in this passage

XII-B@1#eq:tracedensity · line 635
$ℓ_{3} [ψ η] = 3 ψ \partial_{μ} ψ \partial^{μ} ψ .$
XII-B@1#eq:traceEuler · line 641
$EL_{ψ} (ℓ_{3} [ψ η]) = - 3 \partial_{μ} ψ \partial^{μ} ψ - 6 ψ □ ψ = η_{μν} J_{3}^{μν} [ψ η] .$
unlabeled · line 651
$- q R [q] = - 6 e^{2 w} (□ w + \partial_{μ} w \partial^{μ} w) .$
unlabeled · line 657
$- 3 κ □ ψ + \frac{3 κ ^{2}}{2 ( 1 + κ ψ )} \partial_{μ} ψ \partial^{μ} ψ .$
unlabeled · line 663
$- \frac{2}{κ ^{2}} - q R [q] = \frac{6}{κ} □ ψ - \frac{3}{1 + κ ψ} \partial_{μ} ψ \partial^{μ} ψ .$
unlabeled · line 672
$3 (\partial ψ)^{2} - \partial_{μ} (6 ψ \partial^{μ} ψ) = - 3 (\partial ψ)^{2} - 6 ψ □ ψ .$

Lemma: Pure-trace cubic density

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