Manuscript III-B@1 · III-B-v1.tex · lemma · d2c98f62

Lemma: Section identities

The retained first-order jet and its retarded section › A named past-trivial class · lines 918–941 · proof included or linked

III-B@1#lem:section

For $ξ \in G_{R}$ ,

w_{R} (h + K ξ) = w_{R} (h) + ξ^{♯}, D Π_{R} h = 0,

w_{R} (Π_{R} h) = 0, Π_{R}^{2} h = Π_{R} h, G^{(1)} (Π_{R} h) = G^{(1)} (h) .

A homogeneous vector wave in $G_{R}$ is zero.

Proof. Use $D K ξ = □ ξ$ and (eq:inverse). Then

D Π_{R} h = D h - □ R_{□} D h = 0.

The other assertions follow from this equation, $G^{(1)} K = 0$ , and uniqueness for zero wave data.

Displayed formulas in this passage

unlabeled · line 921
$w_{R} (h + K ξ) = w_{R} (h) + ξ^{♯}, D Π_{R} h = 0,$
unlabeled · line 925
$w_{R} (Π_{R} h) = 0, Π_{R}^{2} h = Π_{R} h, G^{(1)} (Π_{R} h) = G^{(1)} (h) .$
unlabeled · line 936
$D Π_{R} h = D h - □ R_{□} D h = 0.$