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

Proposition: Exact first-order defect

Exact first-order repair and the larger-support defect › The defect outside the plateau · lines 1027–1050 · proof included or linked

VII-A@1#prop:defect, VII-A@1#eq:defect

Let $ξ \in C_{c}^{\infty} (M_{+}^{\circ}; T M)$ , without requiring $supp ξ \subset U$ . Then

A_{1}^{(g)} [h + K ξ] - A_{1}^{(g)} [h] = E_{m} Q_{(1 - g) K ξ}^{(m)} ϕ_{0} .

Proof. The coordinate variation is still $ξ$ , since the initial-data condition is unchanged. Thus the left-hand side is

- E_{m} Q_{gK ξ}^{(m)} ϕ_{0} - X_{ξ} ϕ_{0} .

By (eq:QKon) and Lemma lem:leftinverse,

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

Linearity of $Q$ in its tensor argument gives (eq:defect).

Displayed formulas in this passage

VII-A@1#eq:defect · line 1031
$A_{1}^{(g)} [h + K ξ] - A_{1}^{(g)} [h] = E_{m} Q_{(1 - g) K ξ}^{(m)} ϕ_{0} .$
unlabeled · line 1042
$- E_{m} Q_{gK ξ}^{(m)} ϕ_{0} - X_{ξ} ϕ_{0} .$
unlabeled · line 1046
$X_{ξ} ϕ_{0} = - E_{m} Q_{K ξ}^{(m)} ϕ_{0} .$