Manuscript III@2 · III.tex · section · 82dad3e5

The changed scalar equations

The coupled first-order jet and retarded harmonic section › The changed scalar equations · lines 1038–1068

III@2#eq:operator-variation, III@2#eq:jet-scalar

For $g = η + κh$ ,

g^{μν} = η^{μν} - κ h^{μν} (mod κ^{2}),

∣ det g ∣ = 1 + \frac{κ}{2} tr_{η} h (mod κ^{2}) .

Substitution into

□_{g} u = ∣ det g ∣^{- 1/2} \partial_{μ} (∣ det g ∣^{1/2} g^{μν} \partial_{ν} u)

gives

□_{g} - m^{2} = P + κ L_{h} (mod κ^{2}) .

Consequently $ϕ = ϕ_{0} + κ ϕ_{1}$ obeys the coefficient equations

P ϕ_{0} = 0, P ϕ_{1} + L_{h} ϕ_{0} = 0.

The second scalar coefficient is forced, not independently free.

Displayed formulas in this passage

III@2#eq:operator-variation · line 1057
$□_{g} - m^{2} = P + κ L_{h} (mod κ^{2}) .$
III@2#eq:jet-scalar · line 1063
$P ϕ_{0} = 0, P ϕ_{1} + L_{h} ϕ_{0} = 0.$
unlabeled · line 1041
$g^{μν} = η^{μν} - κ h^{μν} (mod κ^{2}),$
unlabeled · line 1044
$∣ det g ∣ = 1 + \frac{κ}{2} tr_{η} h (mod κ^{2}) .$
unlabeled · line 1049
$□_{g} u = ∣ det g ∣^{- 1/2} \partial_{μ} (∣ det g ∣^{1/2} g^{μν} \partial_{ν} u)$