Manuscript X-F@1 · X-F.tex · lemma · f35cc129

Lemma: Flat gauge-fixing derivative

F2: the gauge-fixed kinetic operator and the dynamical jet law › The flat gauge-fixing derivative · lines 725–744 · proof included or linked

X-F@1#lem:F0, X-F@1#eq:F0

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F^{(0)} u = - \frac{1}{4} P K C [u], \frac{1}{2} G_{lin} [u] + F^{(0)} u = P_{gr} u = - \frac{1}{4} P □ u, C^{μ} [K ξ] = □ ξ^{μ} .

Proof. $δ S^{(0)}_{gf} = \frac{1}{2} \int C_{μ} \partial_{ν} (P δ u)^{ν μ} = - \frac{1}{2} \int \partial_{(ν} C_{μ)} (P δ u)^{ν μ} = - \frac{1}{4} \int (P K C)^{ν μ} δ u_{ν μ}$ , using the symmetry of $P$ . With (eq:xe-gf), $\frac{1}{2} G_{lin} [u] = P (- \frac{1}{4} □ u + \frac{1}{4} K C [u])$ , and the $K C$ terms cancel. Finally $(P K ξ)^{ν μ} = \partial^{ν} ξ^{μ} + \partial^{μ} ξ^{ν} - η^{ν μ} \partial \cdot ξ$ , whose divergence is $□ ξ^{μ}$ .

Displayed formulas in this passage

X-F@1#eq:F0 · line 727
$F^{(0)} u = - \frac{1}{4} P K C [u], \frac{1}{2} G_{lin} [u] + F^{(0)} u = P_{gr} u = - \frac{1}{4} P □ u, C^{μ} [K ξ] = □ ξ^{μ} .$

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section: The flat gauge-fixing derivative