Manuscript XI-E@1 · XI-E.tex · section · 46e55514

Variation from tensor-density covariance

Route 3: the cubic Einstein tree and its computed variation › Variation from tensor-density covariance · lines 1574–1592

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The source identity is compatible with XI-A's nonlinear Noether identity. If

Q_{ν} (v) = 2 \partial_{μ} (v^{μ ρ} γ_{ρ ν}) - v^{μ ρ} \partial_{ν} γ_{μ ρ},

its gravitational order-one coefficient is

2 \partial_{μ} (J_{3}^{μ ρ} η_{ρ ν}) + Q_{ν} (A γ) = 0.

For the invariant coupled action, the switching breaking vanishes on $U$ . Gauge fixing nevertheless has its separate contribution, whose leading divergence is $- □ D^{ν}$ . Neither fact permits replacing the full variation by zero.

Displayed formulas in this passage

unlabeled · line 1578
$Q_{ν} (v) = 2 \partial_{μ} (v^{μ ρ} γ_{ρ ν}) - v^{μ ρ} \partial_{ν} γ_{μ ρ},$
unlabeled · line 1584
$2 \partial_{μ} (J_{3}^{μ ρ} η_{ρ ν}) + Q_{ν} (A γ) = 0.$

Nearby in this section

section: Variation from tensor-density covariance
theorem: Theorem: Cubic source and tree variation