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

Lemma: Transformation of the coordinate coefficients

Finite-time harmonic coordinates and second-order descent › Coordinate covariance · lines 1012–1043 · proof included or linked

VI@1#lem:coordinate-transform, VI@1#eq:coordinate-covariance, VI@1#eq:w-transform, VI@1#eq:v-transform

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For the transformations in Proposition prop:gauge-action,

X_{γ^{'}} = F_{ξ, ζ}^{*} X_{γ} mod κ^{3} .

Consequently,

w^{' μ} v^{' μ} = w^{μ} + ξ^{μ}, = v^{μ} + ξ^{ν} \partial_{ν} w^{μ} + ζ^{μ} + \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} .

Proof. Scalar covariance gives

□_{F^{*} γ} (F^{*} X_{γ}^{μ}) = F^{*} (□_{γ} X_{γ}^{μ}) = 0 mod κ^{3} .

Because $F$ is the identity on a neighborhood of the initial surface, $F^{*} X_{γ}^{μ}$ has exactly the data in (eq:harmonic-coordinate-data). Uniqueness in Proposition prop:coordinates proves (eq:coordinate-covariance). Expanding the pullback gives (eq:w-transform)–(eq:v-transform).

Displayed formulas in this passage

VI@1#eq:coordinate-covariance · line 1015
$X_{γ^{'}} = F_{ξ, ζ}^{*} X_{γ} mod κ^{3} .$
VI@1#eq:w-transform · line 1020
$w^{' μ} v^{' μ} = w^{μ} + ξ^{μ}, = v^{μ} + ξ^{ν} \partial_{ν} w^{μ} + ζ^{μ} + \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} .$
VI@1#eq:v-transform · line 1020
$w^{' μ} v^{' μ} = w^{μ} + ξ^{μ}, = v^{μ} + ξ^{ν} \partial_{ν} w^{μ} + ζ^{μ} + \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} .$
unlabeled · line 1032
$□_{F^{*} γ} (F^{*} X_{γ}^{μ}) = F^{*} (□_{γ} X_{γ}^{μ}) = 0 mod κ^{3} .$