Manuscript X-A-ST@1 · X-A-ST.tex · theorem · d9ccbde1

Theorem: Coordinate intertwiner

Relational expansion and covariance: Route I › Naturality and plateau covariance · lines 1113–1136 · proof included or linked

X-A-ST@1#thm:coordinate, X-A-ST@1#eq:intertwiner, X-A-ST@1#eq:coordinatelaws

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For every smooth metric jet and every $F \in G_{p}$ ,

Co_{F^{*} q} = F^{*} Co_{q} = Co_{q} \circ F (mod κ^{3}) .

In coefficients,

w [h^{'}] v [h^{'}, k^{'}] = w [h] + ξ, = v [h, k] + ξ^{ν} \partial_{ν} w [h] + ζ + \frac{1}{2} ξ^{ν} \partial_{ν} ξ .

Proof. By Lemma lem:plateaucoord, each component of $F^{*} Co_{q}$ solves the transformed coordinate problem. It has the same initial data because $F = id$ near $Σ$ . Proposition prop:coordeq gives uniqueness. Expansion of $Co_{q} \circ F$ gives (eq:coordinatelaws).

Displayed formulas in this passage

X-A-ST@1#eq:intertwiner · line 1116
$Co_{F^{*} q} = F^{*} Co_{q} = Co_{q} \circ F (mod κ^{3}) .$
X-A-ST@1#eq:coordinatelaws · line 1121
$w [h^{'}] v [h^{'}, k^{'}] = w [h] + ξ, = v [h, k] + ξ^{ν} \partial_{ν} w [h] + ζ + \frac{1}{2} ξ^{ν} \partial_{ν} ξ .$

Theorem: Coordinate intertwiner

Displayed formulas in this passage

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