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

Lemma: Complete scalar expansion

The second-order Einstein–scalar coefficient space › Normalization and the scalar second variation · lines 540–586 · proof included or linked

VI-B@1#lem:scalar-expansion, VI-B@1#eq:M

One has

□_{γ} - m^{2} = P + κ L_{h} + κ^{2} (L_{j} + M_{h}) (mod κ^{3}),

where

M_{h} u = \partial_{μ} (C_{h}^{μν} \partial_{ν} u) - d_{h} L_{h} u - e_{h} □ u .

Proof. Matrix inversion and the determinant expansion give

γ^{μν} = η^{μν} - κ h^{μν} + κ^{2} (h^{μ}_{ρ} h^{ρ ν} - j^{μν}),

- γ = 1 + κ d_{h} + κ^{2} (d_{j} + e_{h}) .

Their product is

- γ γ^{μν} = η^{μν} + κ A_{h}^{μν} + κ^{2} (A_{j}^{μν} + C_{h}^{μν}) .

Insert this and the inverse density in

□_{γ} u = (- γ)^{- 1} \partial_{μ} (- γ γ^{μν} \partial_{ν} u) .

The linear term is $L_{h}$ , the part linear in $j$ is $L_{j}$ , and the remaining quadratic term is

\partial_{μ} (C_{h}^{μν} \partial_{ν} u) - d_{h} \partial_{μ} (A_{h}^{μν} \partial_{ν} u) + (d_{h}^{2} - e_{h}) □ u,

which equals (eq:M).

Displayed formulas in this passage

VI-B@1#eq:M · line 548
$M_{h} u = \partial_{μ} (C_{h}^{μν} \partial_{ν} u) - d_{h} L_{h} u - e_{h} □ u .$
unlabeled · line 543
$□_{γ} - m^{2} = P + κ L_{h} + κ^{2} (L_{j} + M_{h}) (mod κ^{3}),$
unlabeled · line 558
$γ^{μν} = η^{μν} - κ h^{μν} + κ^{2} (h^{μ}_{ρ} h^{ρ ν} - j^{μν}),$
unlabeled · line 563
$- γ = 1 + κ d_{h} + κ^{2} (d_{j} + e_{h}) .$
unlabeled · line 567
$- γ γ^{μν} = η^{μν} + κ A_{h}^{μν} + κ^{2} (A_{j}^{μν} + C_{h}^{μν}) .$
unlabeled · line 573
$□_{γ} u = (- γ)^{- 1} \partial_{μ} (- γ γ^{μν} \partial_{ν} u) .$
unlabeled · line 580
$\partial_{μ} (C_{h}^{μν} \partial_{ν} u) - d_{h} \partial_{μ} (A_{h}^{μν} \partial_{ν} u) + (d_{h}^{2} - e_{h}) □ u,$