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

Lemma: Coordinate identity

Exact first-order repair and the larger-support defect › Two linear identities · lines 905–930 · proof included or linked

VII-A@1#lem:CK, VII-A@1#eq:CK, VII-A@1#eq:wK

For every smooth vector field $ξ$ ,

C^{μ} [K ξ] = □_{η} ξ^{μ} .

If $ξ$ vanishes near $Σ_{0}$ , then

w [K ξ] = ξ .

Proof. The trace of $K ξ$ is $2 \partial_{ρ} ξ^{ρ}$ , so

\partial_{ν} (K ξ)^{ν μ} = □_{η} ξ^{μ} + \partial^{μ} \partial_{ρ} ξ^{ρ} .

The second term is cancelled by half the derivative of the trace. Equation (eq:wK) then follows from Lemma lem:leftinverse, componentwise.

Displayed formulas in this passage

VII-A@1#eq:CK · line 908
$C^{μ} [K ξ] = □_{η} ξ^{μ} .$
VII-A@1#eq:wK · line 913
$w [K ξ] = ξ .$
unlabeled · line 921
$\partial_{ν} (K ξ)^{ν μ} = □_{η} ξ^{μ} + \partial^{μ} \partial_{ρ} ξ^{ρ} .$