Manuscript XVI-D@1 · XVI-D.tex · proposition · b73c6f4a

Proposition: Conditional inherited-coordinate compatibility

Coordinate candidate and a restricted auxiliary extension › Updated evidence about the inherited definition · lines 1284–1325 · proof included or linked

XVI-D@1#prop:coordinate-compatibility, XVI-D@1#eq:coordinate-compatible, XVI-D@1#eq:direct-conditional

Back to manuscript

Suppose the inherited application $E_{0}^{R} F_{co}$ is defined and is a smooth solution of

□ E_{0}^{R} F_{co} = F_{co}

with zero initial data. Then

X_{1} [R_{λ}] = - E_{0}^{R} F_{co} = X_{cand} .

Consequently the direct inherited completion on $R_{λ}$ exists and equals

A_{D} (λ; f, g, ϕ) = \int_{M_{T}} f (ϕ_{cand} - X_{cand}^{μ} \partial_{μ} ϕ) d^{4} x .

Proof. The first two expressions in (eq:coordinate-compatible) refer to the inherited definition (eq:completion-legs). By the assumed operator property they solve (eq:boxX). Proposition lem:Xcand and Lemma lem:uniqueness give their equality.

The scalar field is already verified by Proposition lem:scalar. Thus both fields in the integrand of (eq:direct-conditional) are smooth. Multiplication by $f$ gives compact support and a finite integral. Substitution in (eq:Aone) proves the formula.

Displayed formulas in this passage

XVI-D@1#eq:coordinate-compatible · line 1294
$X_{1} [R_{λ}] = - E_{0}^{R} F_{co} = X_{cand} .$
XVI-D@1#eq:direct-conditional · line 1302
$A_{D} (λ; f, g, ϕ) = \int_{M_{T}} f (ϕ_{cand} - X_{cand}^{μ} \partial_{μ} ϕ) d^{4} x .$
unlabeled · line 1289
$□ E_{0}^{R} F_{co} = F_{co}$

Proposition: Conditional inherited-coordinate compatibility

Displayed formulas in this passage

Nearby in this section