Manuscript VI@1 · VI.tex · proposition · b5b60afb

Proposition: Existence and uniqueness of the coordinate jet

Finite-time harmonic coordinates and second-order descent › Coordinate coefficients · lines 951–980 · proof included or linked

VI@1#prop:coordinates, VI@1#eq:w, VI@1#eq:v

For every smooth $h, j$ , the conditions (eq:harmonic-coordinate-data) have the unique coefficient solution

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]) .

Both coefficients and their first time derivatives vanish at $t = 0$ .

Proof. Since $□_{η} x^{μ} = 0$ and $L_{h} x^{μ} = - D (h)^{μ}$ , the first coefficient of $□_{γ} X^{μ}$ is

□_{η} w^{μ} - D (h)^{μ} .

The second coefficient is

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

The zero-data wave problem gives the formulas and uniqueness.

Displayed formulas in this passage

VI@1#eq:w · line 955
$w^{μ} [h] v^{μ} [h, j] = G_{□, 0} D (h)^{μ}, = G_{□, 0} (D (j)^{μ} - M_{h} x^{μ} - L_{h} w^{μ} [h]) .$
VI@1#eq:v · line 955
$w^{μ} [h] v^{μ} [h, j] = G_{□, 0} D (h)^{μ}, = G_{□, 0} (D (j)^{μ} - M_{h} x^{μ} - L_{h} w^{μ} [h]) .$
unlabeled · line 972
$□_{η} w^{μ} - D (h)^{μ} .$
unlabeled · line 976
$□_{η} v^{μ} + L_{h} w^{μ} - D (j)^{μ} + M_{h} x^{μ} .$