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

A named past-trivial class

The retained first-order jet and its retarded section › A named past-trivial class · lines 882–917

III-B@1#eq:inverse, III-B@1#eq:ret-section

Let $E_{+, r_{*}}$ consist of supported smooth fields on $L$ that vanish for $s \leq 0$ . For $F$ in this class, let $R_{□} F$ be the zero-data solution of

□ R_{□} F = F

at $s = 0$ , continued as zero for $s < 0$ . Lemma lem:evolution proves existence, smoothness, and preservation of the support class.

For $a \in E_{+, r_{*}}$ ,

□ R_{□} F = F, R_{□} □ a = a .

The second identity follows by homogeneous-wave uniqueness.

Define

G_{R} = E_{+, r_{*}} (T^{*}),

M_{R} = {h : supp h \subseteq C_{r_{*}}, D h = 0 for s \leq 0},

and

w_{R} (h) = R_{□} (D h)^{♯}, Π_{R} h = h - K (w_{R} (h)^{♭}) .

Displayed formulas in this passage

III-B@1#eq:inverse · line 896
$□ R_{□} F = F, R_{□} □ a = a .$
III-B@1#eq:ret-section · line 912
$w_{R} (h) = R_{□} (D h)^{♯}, Π_{R} h = h - K (w_{R} (h)^{♭}) .$
unlabeled · line 888
$□ R_{□} F = F$
unlabeled · line 903
$G_{R} = E_{+, r_{*}} (T^{*}),$
unlabeled · line 906
$M_{R} = {h : supp h \subseteq C_{r_{*}}, D h = 0 for s \leq 0},$

A named past-trivial class

Displayed formulas in this passage

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