Manuscript X-A-conv@1 · X-A-conv.tex · section · c76fffa3

A genuine pure-gauge jet

The complete second-order pure-gauge test › A genuine pure-gauge jet · lines 1872–1917

X-A-conv@1#eq:puregauge, X-A-conv@1#eq:puremetriccoeff, X-A-conv@1#eq:purescalarcoeff, X-A-conv@1#eq:differencejet

For $ξ \in g_{U}$ and a real parameter $λ$ , take

q_{λ} = exp (λκ L_{ξ}) η (mod κ^{3}), ϕ_{λ} = exp (λκ X_{ξ}) ϕ_{0} (mod κ^{3}) .

Thus

h_{λ} ϕ_{λ} = λ K ξ, = ϕ_{0} + λκ X_{ξ} ϕ_{0} + \frac{λ ^{2} κ ^{2}}{2} X_{ξ}^{2} ϕ_{0} . k_{λ} = \frac{λ ^{2}}{2} L_{ξ}^{2} η,

The straight-line metric $η + λκ K ξ$ is not substituted for (eq:puregauge).

Let

a = S_{ε} ξ, d_{ξ} = \frac{1}{2} T_{g, ε} (L_{ξ}^{2} η) - \frac{1}{2} L_{a}^{2} η .

Then the regulated auxiliary jet is

q_{λ} = η + λκ K a + λ^{2} κ^{2} (\frac{1}{2} L_{a}^{2} η + d_{ξ}) .

Displayed formulas in this passage

X-A-conv@1#eq:puregauge · line 1875
$q_{λ} = exp (λκ L_{ξ}) η (mod κ^{3}), ϕ_{λ} = exp (λκ X_{ξ}) ϕ_{0} (mod κ^{3}) .$
X-A-conv@1#eq:puremetriccoeff · line 1886
$h_{λ} ϕ_{λ} = λ K ξ, = ϕ_{0} + λκ X_{ξ} ϕ_{0} + \frac{λ ^{2} κ ^{2}}{2} X_{ξ}^{2} ϕ_{0} . k_{λ} = \frac{λ ^{2}}{2} L_{ξ}^{2} η,$
X-A-conv@1#eq:purescalarcoeff · line 1886
$h_{λ} ϕ_{λ} = λ K ξ, = ϕ_{0} + λκ X_{ξ} ϕ_{0} + \frac{λ ^{2} κ ^{2}}{2} X_{ξ}^{2} ϕ_{0} . k_{λ} = \frac{λ ^{2}}{2} L_{ξ}^{2} η,$
X-A-conv@1#eq:differencejet · line 1901
$a = S_{ε} ξ, d_{ξ} = \frac{1}{2} T_{g, ε} (L_{ξ}^{2} η) - \frac{1}{2} L_{a}^{2} η .$
unlabeled · line 1910
$q_{λ} = η + λκ K a + λ^{2} κ^{2} (\frac{1}{2} L_{a}^{2} η + d_{ξ}) .$