Manuscript III@2 · III.tex · lemma · 82dad3e5

Lemma: Retarded section identities

The coupled first-order jet and retarded harmonic section › Past-trivial gauges · lines 1139–1168 · proof included or linked

III@2#lem:retarded-section

For $ξ \in G_{R}$ ,

w_{R} (K ξ) = ξ, w_{R} (h + K ξ) = w_{R} (h) + ξ .

Also,

D Π_{R} h = 0, w_{R} (Π_{R} h) = 0, Π_{R}^{2} h = Π_{R} h,

and

G^{(1)} (Π_{R} h) = G^{(1)} (h) .

Proof. Apply $D K ξ = □_{η} ξ$ and (eq:retarded-inverse). Then

D Π_{R} h = D h - □_{η} R_{□} D h = 0.

The other projector identities follow, and $G^{(1)} K = 0$ proves the final equality.

Displayed formulas in this passage

unlabeled · line 1142
$w_{R} (K ξ) = ξ, w_{R} (h + K ξ) = w_{R} (h) + ξ .$
unlabeled · line 1147
$D Π_{R} h = 0, w_{R} (Π_{R} h) = 0, Π_{R}^{2} h = Π_{R} h,$
unlabeled · line 1153
$G^{(1)} (Π_{R} h) = G^{(1)} (h) .$
unlabeled · line 1162
$D Π_{R} h = D h - □_{η} R_{□} D h = 0.$