Manuscript VI@1 · VI.tex · section · b5b60afb

What linearization at the chosen background gives

The linearized interface and a sourced nonempty domain › What linearization at the chosen background gives · lines 599–628

VI@1#eq:free-operators, VI@1#eq:free-Kstar

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At $(η, 0)$ , the ordinary linearized scalar gauge variation vanishes, because the background scalar has zero derivative. With canonical metric normalization, the free Hessian and gauge map are

P_{0} (a, u) = (- 2 G^{(1)} (a), P u), K_{0} ξ = (K ξ, 0) .

Metric indices in the first entry are raised when it is paired with a covariant variation.

For compact tests $(J, c)$ , use

⟨ (J, c), (a, u) ⟩ = \int J^{μν} a_{μν} d^{4} x + \int c u d^{4} x .

The formal adjoint gauge map is

(K_{0}^{*} (J, c))_{ν} = - 2 η_{ν β} \partial_{μ} J^{μ β} .

Displayed formulas in this passage

VI@1#eq:free-operators · line 604
$P_{0} (a, u) = (- 2 G^{(1)} (a), P u), K_{0} ξ = (K ξ, 0) .$
VI@1#eq:free-Kstar · line 622
$(K_{0}^{*} (J, c))_{ν} = - 2 η_{ν β} \partial_{μ} J^{μ β} .$
unlabeled · line 614
$⟨ (J, c), (a, u) ⟩ = \int J^{μν} a_{μν} d^{4} x + \int c u d^{4} x .$

What linearization at the chosen background gives

Displayed formulas in this passage

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