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

Lemma: Linear identities

The second-order Einstein–scalar coefficient space · lines 454–480 · proof included or linked

VI-B@1#lem:linear, VI-B@1#eq:linear-identities

One has

D (K ξ) = □ ξ, G^{(1)} (K ξ) = 0, \partial^{μ} G^{(1)} (a)_{μν} = 0, L_{K ξ} = - [P, X_{ξ}] .

Proof. The mixed derivatives of $\partial_{μ} ξ^{μ}$ cancel in $D (K ξ)$ . Substituting this result in (eq:G1) proves the pure-gauge identity, and taking a divergence proves the linearized Bianchi identity.

Finally,

[P, X_{ξ}] u = (□ ξ^{ν}) \partial_{ν} u + 2 (\partial^{μ} ξ^{ν}) \partial_{μ} \partial_{ν} u,

which is $- L_{K ξ} u$ .

Displayed formulas in this passage

VI-B@1#eq:linear-identities · line 457
$D (K ξ) = □ ξ, G^{(1)} (K ξ) = 0, \partial^{μ} G^{(1)} (a)_{μν} = 0, L_{K ξ} = - [P, X_{ξ}] .$
unlabeled · line 474
$[P, X_{ξ}] u = (□ ξ^{ν}) \partial_{ν} u + 2 (\partial^{μ} ξ^{ν}) \partial_{μ} \partial_{ν} u,$

Nearby in this section

section: The second-order Einstein–scalar coefficient space