Manuscript XI-A@1 · XI-A.tex · section · b8211ecc

Coefficient and sign audit

Noether identity and the inherited Route II › Coefficient and sign audit · lines 874–901

XI-A@1#eq:N1, XI-A@1#eq:N2

For a $κ$ -independent scalar write $e = e_{0} + κ e_{1} + κ^{2} e_{2} + \dots$ and

Q_{ν} (u) = 2 \partial_{μ} (u^{μ ρ} γ_{ρ ν}) - u^{μ ρ} \partial_{ν} γ_{μ ρ} .

Then (eq:noether) gives

2 \partial_{μ} (e_{0}^{μ ρ} η_{ρ ν}) 2 \partial_{μ} (e_{1}^{μ ρ} η_{ρ ν}) + Q_{ν} (e_{0}) 2 \partial_{μ} (e_{2}^{μ ρ} η_{ρ ν}) + Q_{ν} (e_{1}) = 0, = - g (P_{m} ϕ) \partial_{ν} ϕ - (\partial_{μ} g) T_{0}^{μ}_{ν}, = - (V_{1, g} ϕ) \partial_{ν} ϕ + (\partial_{ν} g) L^{(1)} [γ, ϕ],

where $L^{(1)} = \frac{1}{2} (a_{1}^{α β} p_{α} p_{β} - m^{2} b_{1} ϕ^{2})$ . These formulas expose the missing coupling and nonlinear metric terms in a bare partial-divergence identity. The source sign in (eq:N1) agrees with $- \frac{1}{2} g T_{0}$ in the metric derivative.

Displayed formulas in this passage

XI-A@1#eq:N1 · line 884
$2 \partial_{μ} (e_{0}^{μ ρ} η_{ρ ν}) 2 \partial_{μ} (e_{1}^{μ ρ} η_{ρ ν}) + Q_{ν} (e_{0}) 2 \partial_{μ} (e_{2}^{μ ρ} η_{ρ ν}) + Q_{ν} (e_{1}) = 0, = - g (P_{m} ϕ) \partial_{ν} ϕ - (\partial_{μ} g) T_{0}^{μ}_{ν}, = - (V_{1, g} ϕ) \partial_{ν} ϕ + (\partial_{ν} g) L^{(1)} [γ, ϕ],$
XI-A@1#eq:N2 · line 884
$2 \partial_{μ} (e_{0}^{μ ρ} η_{ρ ν}) 2 \partial_{μ} (e_{1}^{μ ρ} η_{ρ ν}) + Q_{ν} (e_{0}) 2 \partial_{μ} (e_{2}^{μ ρ} η_{ρ ν}) + Q_{ν} (e_{1}) = 0, = - g (P_{m} ϕ) \partial_{ν} ϕ - (\partial_{μ} g) T_{0}^{μ}_{ν}, = - (V_{1, g} ϕ) \partial_{ν} ϕ + (\partial_{ν} g) L^{(1)} [γ, ϕ],$
unlabeled · line 878
$Q_{ν} (u) = 2 \partial_{μ} (u^{μ ρ} γ_{ρ ν}) - u^{μ ρ} \partial_{ν} γ_{μ ρ} .$