Manuscript VI@1 · VI.tex · proposition · b5b60afb

Proposition: Coefficient gauge action

Action, signs, background, and coefficient domain › The named gauge group · lines 529–571 · proof included or linked

VI@1#prop:gauge-action, VI@1#eq:h-transform, VI@1#eq:j-transform, VI@1#eq:phi0-transform, VI@1#eq:phi1-transform, VI@1#eq:phi2-transform

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The action (eq:formal-pullback) induces

h^{'} j^{'} ϕ_{0}^{'} ϕ_{1}^{'} ϕ_{2}^{'} = h + K ξ, = j + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ}^{2} η, = ϕ_{0}, = ϕ_{1} + X_{ξ} ϕ_{0}, = ϕ_{2} + X_{ξ} ϕ_{1} + X_{ζ} ϕ_{0} + \frac{1}{2} X_{ξ}^{2} ϕ_{0} .

It preserves $J_{T}$ and all coefficient Cauchy data at $t = 0$ .

Proof. Expand the exponential in (eq:formal-pullback) and collect powers of $κ$ after substituting (eq:jets). This gives the displayed formulas.

The tensor identity

G (F^{*} γ) = F^{*} G (γ)

and the scalar identity

(□_{F^{*} γ} - m^{2}) (F^{*} ϕ) = F^{*} ((□_{γ} - m^{2}) ϕ)

preserve the coefficient equations. These identities follow from the tensorial definitions of the Einstein tensor and the scalar wave operator. Equivalently, they may be verified by substituting the pullback formulas into their coordinate definitions and applying the chain rule.

Since $ξ, ζ$ vanish on a neighborhood of $t = 0$ , all displayed changes and their time derivatives vanish there.

Displayed formulas in this passage

VI@1#eq:h-transform · line 532
$h^{'} j^{'} ϕ_{0}^{'} ϕ_{1}^{'} ϕ_{2}^{'} = h + K ξ, = j + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ}^{2} η, = ϕ_{0}, = ϕ_{1} + X_{ξ} ϕ_{0}, = ϕ_{2} + X_{ξ} ϕ_{1} + X_{ζ} ϕ_{0} + \frac{1}{2} X_{ξ}^{2} ϕ_{0} .$
VI@1#eq:j-transform · line 532
$h^{'} j^{'} ϕ_{0}^{'} ϕ_{1}^{'} ϕ_{2}^{'} = h + K ξ, = j + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ}^{2} η, = ϕ_{0}, = ϕ_{1} + X_{ξ} ϕ_{0}, = ϕ_{2} + X_{ξ} ϕ_{1} + X_{ζ} ϕ_{0} + \frac{1}{2} X_{ξ}^{2} ϕ_{0} .$
VI@1#eq:phi0-transform · line 532
$h^{'} j^{'} ϕ_{0}^{'} ϕ_{1}^{'} ϕ_{2}^{'} = h + K ξ, = j + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ}^{2} η, = ϕ_{0}, = ϕ_{1} + X_{ξ} ϕ_{0}, = ϕ_{2} + X_{ξ} ϕ_{1} + X_{ζ} ϕ_{0} + \frac{1}{2} X_{ξ}^{2} ϕ_{0} .$
VI@1#eq:phi1-transform · line 532
$h^{'} j^{'} ϕ_{0}^{'} ϕ_{1}^{'} ϕ_{2}^{'} = h + K ξ, = j + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ}^{2} η, = ϕ_{0}, = ϕ_{1} + X_{ξ} ϕ_{0}, = ϕ_{2} + X_{ξ} ϕ_{1} + X_{ζ} ϕ_{0} + \frac{1}{2} X_{ξ}^{2} ϕ_{0} .$
VI@1#eq:phi2-transform · line 532
$h^{'} j^{'} ϕ_{0}^{'} ϕ_{1}^{'} ϕ_{2}^{'} = h + K ξ, = j + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ}^{2} η, = ϕ_{0}, = ϕ_{1} + X_{ξ} ϕ_{0}, = ϕ_{2} + X_{ξ} ϕ_{1} + X_{ζ} ϕ_{0} + \frac{1}{2} X_{ξ}^{2} ϕ_{0} .$
unlabeled · line 554
$G (F^{*} γ) = F^{*} G (γ)$
unlabeled · line 558
$(□_{F^{*} γ} - m^{2}) (F^{*} ϕ) = F^{*} ((□_{γ} - m^{2}) ϕ)$

Proposition: Coefficient gauge action

Displayed formulas in this passage

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