Manuscript V-B@1 · V-B-v1.tex · lemma · 3fadc1ad

Lemma: Differential identities

Operators and the fixed-time response · lines 348–372 · proof included or linked

V-B@1#lem:identities

The following identities hold:

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

The operator $G^{(1)}$ is formally self-adjoint.

Proof. In $D K ξ$ , the derivatives of $\partial_{μ} ξ^{μ}$ cancel. Substitution in (eq:Einstein) proves $G^{(1)} K = 0$ , and taking a divergence proves the Bianchi identity. Twice applying the product rule gives

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

which is $- L_{K ξ} ϕ$ . Integration by parts in the constant-coefficient expression (eq:Einstein) proves formal self-adjointness.

Displayed formulas in this passage

unlabeled · line 351
$D K = □, G^{(1)} K = 0, \partial^{μ} G^{(1)} (h)_{μν} = 0, L_{K ξ} = - [P, X_{ξ}] .$
unlabeled · line 364
$[P, X_{ξ}] ϕ = (□ ξ^{ν}) \partial_{ν} ϕ + 2 (\partial^{μ} ξ^{ν}) \partial_{μ} \partial_{ν} ϕ,$

Lemma: Differential identities

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