Manuscript Em06-B@1 · Em06-B-v1.tex · proposition · 61e38859

Proposition: Plateau gauge identity, before and after scalar descent

Independent classical audit and compact-test estimates · lines 436–476 · proof included or linked

Em06-B@1#prop:gauge, Em06-B@1#eq:plateau, Em06-B@1#eq:test-gauge

For $ξ \in g_{U}$ ,

w [h + K ξ] - w [h] ϕ_{1, χ} [h + K ξ; ϕ_{0}] - ϕ_{1, χ} [h; ϕ_{0}] O_{f, h + K ξ, χ} [ϕ_{0}] = ξ, = X_{ξ} ϕ_{0}, = O_{f, h, χ} [ϕ_{0}] .

At the test level,

q_{f, h + K ξ, χ} - q_{f, h, χ} = P_{m} (X_{ξ} u_{f}), X_{ξ} u_{f} \in C_{c}^{\infty} (M) .

Proof. Since $ξ$ and all its derivatives are supported where $χ = 1$ , $χK ξ = K ξ$ . On the free scalar shell,

Q_{K ξ} ϕ_{0} = - P_{m} (X_{ξ} ϕ_{0}) .

Retarded inversion gives $G_{m}^{R} P_{m} (X_{ξ} ϕ_{0}) = X_{ξ} ϕ_{0}$ , because this smooth function is compactly supported away from the initial surface. Equation (eq:CK) and Cauchy uniqueness similarly give $w [K ξ] = ξ$ . The two variations in (eq:observable) cancel.

For the stronger compact-test identity, $P_{m} u_{f} = f$ and (eq:QK) yield

- Q_{K ξ} u_{f} + \partial_{μ} (f ξ^{μ}) = P_{m} (X_{ξ} u_{f}) .

The multiplier $ξ$ makes $X_{ξ} u_{f}$ compact.

Displayed formulas in this passage

Em06-B@1#eq:plateau · line 439
$w [h + K ξ] - w [h] ϕ_{1, χ} [h + K ξ; ϕ_{0}] - ϕ_{1, χ} [h; ϕ_{0}] O_{f, h + K ξ, χ} [ϕ_{0}] = ξ, = X_{ξ} ϕ_{0}, = O_{f, h, χ} [ϕ_{0}] .$
Em06-B@1#eq:test-gauge · line 448
$q_{f, h + K ξ, χ} - q_{f, h, χ} = P_{m} (X_{ξ} u_{f}), X_{ξ} u_{f} \in C_{c}^{\infty} (M) .$
unlabeled · line 460
$Q_{K ξ} ϕ_{0} = - P_{m} (X_{ξ} ϕ_{0}) .$
unlabeled · line 471
$- Q_{K ξ} u_{f} + \partial_{μ} (f ξ^{μ}) = P_{m} (X_{ξ} u_{f}) .$

Proposition: Plateau gauge identity, before and after scalar descent

Displayed formulas in this passage

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