Manuscript X-E@1 · X-E.tex · lemma · 293fae6a

Lemma: De Donder violation of a retarded response

E3: the quadratic operator, its inverse, and the residual algebra › De Donder gauge fixing and the operator · lines 1080–1094 · proof included or linked

X-E@1#lem:violation, X-E@1#eq:violation

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For a smooth symmetric source $S^{μν}$ vanishing on $t \leq 0$ ,

C^{μ} [Δ_{R}^{gr} S] = - 4 E_{0}^{R} (\partial_{ν} S^{ν μ}) .

In particular $Δ_{R}^{gr} S$ is in de Donder gauge iff $S$ is conserved.

Proof. $P Δ_{R}^{gr} S = - 4 E_{0}^{R} S$ , so $C^{μ} = \partial_{ν} (P Δ_{R}^{gr} S)^{ν μ} = - 4 \partial_{ν} E_{0}^{R} S^{ν μ}$ , and $E_{0}^{R}$ commutes with constant-coefficient derivatives.

Displayed formulas in this passage

X-E@1#eq:violation · line 1083
$C^{μ} [Δ_{R}^{gr} S] = - 4 E_{0}^{R} (\partial_{ν} S^{ν μ}) .$

Nearby in this section

definition: Definition
lemma: Lemma: Gauge-fixed operator