Manuscript VII-A@1 · VII-A.tex · section · 11da3fe6

A pure-gauge metric jet

Consistency checks › A pure-gauge metric jet · lines 1791–1818

VII-A@1#eq:puregaugejet, VII-A@1#eq:purescalar, VII-A@1#eq:purecoordinates

Start from the flat metric and take $ζ = 0$ . Its pullback has

h = K ξ, k = \frac{1}{2} L_{ξ}^{2} η .

For $ξ \in g_{U}$ , the intertwining formulas give

ϕ_{1, g} w^{μ} = X_{ξ} ϕ_{0}, = ξ^{μ}, ϕ_{2, g} v^{μ} = \frac{1}{2} X_{ξ}^{2} ϕ_{0}, = \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} .

Substitution into (eq:A1)–(eq:A2) yields

A_{1}^{(g)} = A_{2}^{(g)} = 0.

This check uses the second-order metric coefficient in (eq:puregaugejet). The straight-line family $η + κ K ξ$ with $k = 0$ is generally not a pure-gauge family through second order.

Displayed formulas in this passage

VII-A@1#eq:puregaugejet · line 1794
$h = K ξ, k = \frac{1}{2} L_{ξ}^{2} η .$
VII-A@1#eq:purescalar · line 1800
$ϕ_{1, g} w^{μ} = X_{ξ} ϕ_{0}, = ξ^{μ}, ϕ_{2, g} v^{μ} = \frac{1}{2} X_{ξ}^{2} ϕ_{0}, = \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} .$
VII-A@1#eq:purecoordinates · line 1800
$ϕ_{1, g} w^{μ} = X_{ξ} ϕ_{0}, = ξ^{μ}, ϕ_{2, g} v^{μ} = \frac{1}{2} X_{ξ}^{2} ϕ_{0}, = \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} .$
unlabeled · line 1811
$A_{1}^{(g)} = A_{2}^{(g)} = 0.$