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

The response divergence

D1: the smooth classical response › The response divergence · lines 534–568

XII-D@1#eq:source-divergence, XII-D@1#eq:ramp-defect

We record the relevant XII-C identity before using it. For a symmetric contravariant density $v$ , define

N_{0, ν} (v) = 2 η_{ρ ν} \partial_{μ} v^{μ ρ}, Q_{ν} (v) = 2 \partial_{μ} (v^{μ ρ} γ_{ρ ν}) - v^{μ ρ} \partial_{ν} γ_{μ ρ} .

The actual source obeys

N_{0, ν} (J_{D}) = - σ_{ε} Q_{ν} (A γ) + R_{σ_{ε}}_{ν} [γ],

where

R_{σ_{ε}}_{ν} = 2 η_{ρ ν} (\partial_{μ} σ_{ε}) J_{3}^{μ ρ} + 4 η_{ρ ν} [(\partial_{μ} C_{2}^{μ ρ α β}) \partial_{α} \partial_{β} σ_{ε} - (\partial_{μ} M_{2}^{μ ρ λ}) \partial_{λ} σ_{ε} - M_{2}^{μ ρ λ} \partial_{μ} \partial_{λ} σ_{ε}] .

The apparent third ramp derivative cancels in XII-C's coefficient identity. No conservation assumption is made.

Displayed formulas in this passage

XII-D@1#eq:source-divergence · line 546
$N_{0, ν} (J_{D}) = - σ_{ε} Q_{ν} (A γ) + R_{σ_{ε}}_{ν} [γ],$
XII-D@1#eq:ramp-defect · line 552
$R_{σ_{ε}}_{ν} = 2 η_{ρ ν} (\partial_{μ} σ_{ε}) J_{3}^{μ ρ} + 4 η_{ρ ν} [(\partial_{μ} C_{2}^{μ ρ α β}) \partial_{α} \partial_{β} σ_{ε} - (\partial_{μ} M_{2}^{μ ρ λ}) \partial_{λ} σ_{ε} - M_{2}^{μ ρ λ} \partial_{μ} \partial_{λ} σ_{ε}] .$
unlabeled · line 538
$N_{0, ν} (v) = 2 η_{ρ ν} \partial_{μ} v^{μ ρ}, Q_{ν} (v) = 2 \partial_{μ} (v^{μ ρ} γ_{ρ ν}) - v^{μ ρ} \partial_{ν} γ_{μ ρ} .$