Manuscript VII-B@1 · VII-B.tex · theorem · be632835

Theorem: Linear on-shell repair

Two precise repairs and their limitations › A linear on-shell spacetime representative · lines 1171–1226 · proof included or linked

VII-B@1#thm:onshell, VII-B@1#eq:on-shell-relation, VII-B@1#eq:on-shell-equality, VII-B@1#eq:time-slice-representative

Back to manuscript

Let $q = q_{a} = L^{*} a$ . There exist a smooth compactly supported tensor $q_{sm}$ and a compactly supported tensor distribution $r$ such that

q - q_{sm} = □ r .

Consequently,

⟨ q, h ⟩ = ⟨ q_{sm}, h ⟩

for every smooth solution of $□ h = 0$ .

A construction is obtained by setting $u = E q$ , choosing a smooth time function $ρ (t)$ equal to zero sufficiently far in the past and one sufficiently far in the future, and defining

q_{sm} = [□, ρ] u = ρ^{''} u + 2 ρ^{'} \partial_{t} u .

Proof. Lemma lem:causal-smoothing gives a smooth, spacelike-compact homogeneous solution $u$ . The derivatives of $ρ$ have compact time support. Therefore $q_{sm}$ is smooth and compactly supported.

The identities

E_{R} q_{sm} = ρ u, E_{A} q_{sm} = (ρ - 1) u

follow from the wave equation and uniqueness with zero data in the appropriate time direction. Thus

E q_{sm} = u = E q .

Set $v = q - q_{sm}$ . Then $v$ is compactly supported and $E v = 0$ , so

r := E_{R} v = E_{A} v .

Its support is contained in

J^{+} (supp v) \cap J^{-} (supp v),

a compact set. Moreover, $□ r = v$ , proving (eq:on-shell-relation). Pairing that identity with a smooth solution $h$ and using the formal self-adjointness of $□$ proves (eq:on-shell-equality).

Displayed formulas in this passage

VII-B@1#eq:on-shell-relation · line 1176
$q - q_{sm} = □ r .$
VII-B@1#eq:on-shell-equality · line 1181
$⟨ q, h ⟩ = ⟨ q_{sm}, h ⟩$
VII-B@1#eq:time-slice-representative · line 1190
$q_{sm} = [□, ρ] u = ρ^{''} u + 2 ρ^{'} \partial_{t} u .$
unlabeled · line 1204
$E_{R} q_{sm} = ρ u, E_{A} q_{sm} = (ρ - 1) u$
unlabeled · line 1210
$E q_{sm} = u = E q .$
unlabeled · line 1215
$r := E_{R} v = E_{A} v .$
unlabeled · line 1219
$J^{+} (supp v) \cap J^{-} (supp v),$

Theorem: Linear on-shell repair

Displayed formulas in this passage

Nearby in this section