Manuscript VII-A@1 · VII-A.tex · lemma · 11da3fe6

Lemma: Existence and uniqueness of the coordinate jet

Unswitched harmonic coordinates › Definition and boundary prescription · lines 782–807 · proof included or linked

VII-A@1#lem:coordinateunique

Equations (eq:w)–(eq:v) uniquely solve (eq:harmonic)–(eq:harmonicdata) through second order.

Proof. At first order,

□_{η} w^{μ} = - \partial_{α} A^{α μ} (h) = C^{μ} [h] .

At second order,

□_{η} v^{μ} = C^{μ} [k] - \partial_{α} B^{α μ} (h) - Q_{h}^{(0)} w^{μ} .

Both coefficients have zero Cauchy data. Existence and uniqueness follow as in Lemma lem:scalarunique.

Displayed formulas in this passage

unlabeled · line 790
$□_{η} w^{μ} = - \partial_{α} A^{α μ} (h) = C^{μ} [h] .$
unlabeled · line 798
$□_{η} v^{μ} = C^{μ} [k] - \partial_{α} B^{α μ} (h) - Q_{h}^{(0)} w^{μ} .$