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

Lemma: Coordinate covariance

Finite-time harmonic coordinates and descent · lines 724–746 · proof included or linked

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

Under (eq:gauge),

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

In particular,

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 shows that $F^{*} X_{γ}^{μ}$ solves the harmonic equation for $F^{*} γ$ . Because $F$ is the identity near $Σ_{0}$ , it has precisely the data (eq:coordinate-data). Coefficientwise uniqueness in Proposition prop:coordinates identifies the solutions. Expansion gives (eq:coordinate-transform).

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VI-B@1#eq:coordinate-transform · line 731
$w^{' μ} v^{' μ} = w^{μ} + ξ^{μ}, = v^{μ} + ξ^{ν} \partial_{ν} w^{μ} + ζ^{μ} + \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} .$
unlabeled · line 727
$X_{γ^{'}} = F_{ξ, ζ}^{*} X_{γ} (mod κ^{3}) .$

Lemma: Coordinate covariance

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