Manuscript XIII-A@1 · XIII-A.tex · corollary · f56e91c7

Corollary: Properness of the classical subclass

A3: the coefficientwise restriction kernel › Compact smearings detecting every nonempty region · lines 1106–1127 · proof included or linked

XIII-A@1#cor:proper, XIII-A@1#eq:ordering-solution

For every nonempty $W$ , $C_{W} ⊊ C_{sol}$ . Moreover the constraint smearings used above are not elements of $N_{sol}$ .

Proof. Using the solution $χ$ in Lemma lem:compact-pair, take

u = (η χ, 0, 0, 0, - \partial χ) .

The metric is a wave solution and $D (η χ) = - \partial χ$ , so $u \in C_{sol}$ . Since $□ χ = 0$ ,

(D^{*} u_{b})_{μν} = \partial_{μ} \partial_{ν} χ, L_{q} (u) = C (q, h) > 0.

Thus $u \in / C_{W}$ and $L_{q} \in / N_{sol}$ .

Displayed formulas in this passage

XIII-A@1#eq:ordering-solution · line 1115
$u = (η χ, 0, 0, 0, - \partial χ) .$
unlabeled · line 1122
$(D^{*} u_{b})_{μν} = \partial_{μ} \partial_{ν} χ, L_{q} (u) = C (q, h) > 0.$