Manuscript VI-B@1 · VI-B-v1.tex · theorem · 33798723

Theorem: Exact first-order switch defect

Compact switching: exact defect and plateau obstruction · lines 903–951 · proof included or linked

VI-B@1#thm:switch-defect, VI-B@1#eq:defect

For every smooth free scalar $ϕ_{0}$ ,

A_{f, g}^{(1)} (h + K ξ, ϕ_{0}) - A_{f, g}^{(1)} (h, ϕ_{0}) = \int f [(g - 1) X_{ξ} ϕ_{0} + G_{R} ((\partial_{μ} g) K ξ^{μν} \partial_{ν} ϕ_{0} - [P, g] X_{ξ} ϕ_{0})] d^{4} x,

where

[P, g] u = (□ g) u + 2 (\partial^{μ} g) \partial_{μ} u .

Proof. The coordinate variation is

w [h + K ξ] - w [h] = G_{□, 0} □ ξ = ξ,

because $ξ$ vanishes near the initial surface.

By Lemma lem:linear and (eq:Qswitch),

Q_{g K ξ} ϕ_{0} = - g P (X_{ξ} ϕ_{0}) - (\partial_{μ} g) K ξ^{μν} \partial_{ν} ϕ_{0} .

Therefore

δ_{ξ} ϕ_{1, g} = G_{R} (g P (X_{ξ} ϕ_{0}) + (\partial_{μ} g) K ξ^{μν} \partial_{ν} ϕ_{0}) .

Use

g P (X_{ξ} ϕ_{0}) = P (g X_{ξ} ϕ_{0}) - [P, g] X_{ξ} ϕ_{0} .

The function $g X_{ξ} ϕ_{0}$ is compactly supported after $0$ , so zero-past uniqueness gives $G_{R} P (g X_{ξ} ϕ_{0}) = g X_{ξ} ϕ_{0}$ . Subtract $X_{ξ} ϕ_{0}$ , the dressing variation, and smear.

Displayed formulas in this passage

VI-B@1#eq:defect · line 906
$A_{f, g}^{(1)} (h + K ξ, ϕ_{0}) - A_{f, g}^{(1)} (h, ϕ_{0}) = \int f [(g - 1) X_{ξ} ϕ_{0} + G_{R} ((\partial_{μ} g) K ξ^{μν} \partial_{ν} ϕ_{0} - [P, g] X_{ξ} ϕ_{0})] d^{4} x,$
unlabeled · line 918
$[P, g] u = (□ g) u + 2 (\partial^{μ} g) \partial_{μ} u .$
unlabeled · line 925
$w [h + K ξ] - w [h] = G_{□, 0} □ ξ = ξ,$
unlabeled · line 931
$Q_{g K ξ} ϕ_{0} = - g P (X_{ξ} ϕ_{0}) - (\partial_{μ} g) K ξ^{μν} \partial_{ν} ϕ_{0} .$
unlabeled · line 937
$δ_{ξ} ϕ_{1, g} = G_{R} (g P (X_{ξ} ϕ_{0}) + (\partial_{μ} g) K ξ^{μν} \partial_{ν} ϕ_{0}) .$
unlabeled · line 944
$g P (X_{ξ} ϕ_{0}) = P (g X_{ξ} ϕ_{0}) - [P, g] X_{ξ} ϕ_{0} .$

Theorem: Exact first-order switch defect

Displayed formulas in this passage

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