Manuscript VI-B@1 · VI-B-v1.tex · proposition · 33798723

Proposition: Coordinate construction

Finite-time harmonic coordinates and descent · lines 696–722 · proof included or linked

VI-B@1#prop:coordinates, VI-B@1#eq:coordinates

For arbitrary smooth $h, j$ , the unique coefficient solution of (eq:coordinate-data) is

w^{μ} [h] v^{μ} [h, j] = G_{□, 0} D (h)^{μ}, = G_{□, 0} (D (j)^{μ} - M_{h} x^{μ} - L_{h} w^{μ} [h]) .

The coefficients and their first time derivatives vanish at $0$ . They depend continuously on $h, j$ on compact truncated pasts.

Proof. Since $□ x^{μ} = 0$ and $L_{h} x^{μ} = - D (h)^{μ}$ , the first coordinate equation is

□ w^{μ} = D (h)^{μ} .

The second is

□ v^{μ} = D (j)^{μ} - M_{h} x^{μ} - L_{h} w^{μ} .

Lemma lem:wave proves existence, uniqueness, locality, and continuity for their zero-data problems.

Displayed formulas in this passage

VI-B@1#eq:coordinates · line 700
$w^{μ} [h] v^{μ} [h, j] = G_{□, 0} D (h)^{μ}, = G_{□, 0} (D (j)^{μ} - M_{h} x^{μ} - L_{h} w^{μ} [h]) .$
unlabeled · line 713
$□ w^{μ} = D (h)^{μ} .$
unlabeled · line 717
$□ v^{μ} = D (j)^{μ} - M_{h} x^{μ} - L_{h} w^{μ} .$