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

Switch and plateau

Cauchy problems and the admitted gauge domain › Switch and plateau · lines 413–452

VII-A@1#eq:densityLie, VII-A@1#eq:scalarpull, VII-A@1#eq:densitypull, VII-A@1#eq:gfixed

The Lie algebra of formal transformations (eq:formalF) is closed modulo $κ^{3}$ by the Baker–Campbell–Hausdorff formula. In particular, omitting $ζ$ would not give a convenient composition-stable parametrization.

For scalars define

X_{ξ} = ξ^{μ} \partial_{μ} .

For scalar densities, represented relative to $d^{4} x$ , define

D_{ξ} = X_{ξ} + \partial_{μ} ξ^{μ} .

Thus

F_{κ}^{*} ψ F_{κ, dens}^{*} r = ψ + κ X_{ξ} ψ + κ^{2} (X_{ζ} + \frac{1}{2} X_{ξ}^{2}) ψ + O (κ^{3}), = r + κ D_{ξ} r + κ^{2} (D_{ζ} + \frac{1}{2} D_{ξ}^{2}) r + O (κ^{3}) .

Every $F \in G_{U}$ preserves $g$ :

F^{*} g = g .

Indeed, $g = 1$ on $U$ , $F (U) = U$ , and $F$ is the identity elsewhere.

Displayed formulas in this passage

VII-A@1#eq:densityLie · line 425
$D_{ξ} = X_{ξ} + \partial_{μ} ξ^{μ} .$
VII-A@1#eq:scalarpull · line 430
$F_{κ}^{*} ψ F_{κ, dens}^{*} r = ψ + κ X_{ξ} ψ + κ^{2} (X_{ζ} + \frac{1}{2} X_{ξ}^{2}) ψ + O (κ^{3}), = r + κ D_{ξ} r + κ^{2} (D_{ζ} + \frac{1}{2} D_{ξ}^{2}) r + O (κ^{3}) .$
VII-A@1#eq:densitypull · line 430
$F_{κ}^{*} ψ F_{κ, dens}^{*} r = ψ + κ X_{ξ} ψ + κ^{2} (X_{ζ} + \frac{1}{2} X_{ξ}^{2}) ψ + O (κ^{3}), = r + κ D_{ξ} r + κ^{2} (D_{ζ} + \frac{1}{2} D_{ξ}^{2}) r + O (κ^{3}) .$
VII-A@1#eq:gfixed · line 447
$F^{*} g = g .$
unlabeled · line 421
$X_{ξ} = ξ^{μ} \partial_{μ} .$

Switch and plateau

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