Manuscript VII-A-ST@1 · VII-A-ST.tex · section · c39a67a3

Switching before variation

Switching before variation · lines 443–494

VII-A-ST@1#eq:metricjet, VII-A-ST@1#eq:AB, VII-A-ST@1#eq:density-expansion

Work in $C^{\infty} (D) [[κ]] / (κ^{3})$ , without factorials in the coefficient expansion:

γ = η + κh + κ^{2} k (mod κ^{3}) .

The prescribed background $g ˉ$ and the formal unknown $γ$ have different roles. The former supplies the laboratory and apparatus cone. The latter is an input to the coefficient equations. No record-dependent metric update is introduced.

Raise coefficient indices with $η$ . Define

a (h) A^{μν} (h) = \frac{1}{2} tr_{η} h, = a (h) η^{μν} - h^{μν}, b (h) B^{μν} (h) = \frac{1}{8} (tr_{η} h)^{2} - \frac{1}{4} h_{μν} h^{μν}, = h^{μ}_{ρ} h^{ρ ν} - a (h) h^{μν} + b (h) η^{μν} .

Writing

ρ_{γ} = - det γ, A_{γ}^{μν} = ρ_{γ} γ^{μν},

matrix inversion and determinant expansion give

ρ_{γ} A_{γ}^{μν} = 1 + κa (h) + κ^{2} (a (k) + b (h)) (mod κ^{3}), = η^{μν} + κ A^{μν} (h) + κ^{2} (A^{μν} (k) + B^{μν} (h)) (mod κ^{3}) .

Indeed,

γ^{- 1} = η^{- 1} - κ h^{♯} + κ^{2} (h^{♯} h^{♯} - k^{♯}),

and $ρ_{γ} = exp (\frac{1}{2} tr lo g (η^{- 1} γ))$ ; multiplication proves (eq:density-expansion).

Displayed formulas in this passage

VII-A-ST@1#eq:metricjet · line 448
$γ = η + κh + κ^{2} k (mod κ^{3}) .$
VII-A-ST@1#eq:AB · line 459
$a (h) A^{μν} (h) = \frac{1}{2} tr_{η} h, = a (h) η^{μν} - h^{μν}, b (h) B^{μν} (h) = \frac{1}{8} (tr_{η} h)^{2} - \frac{1}{4} h_{μν} h^{μν}, = h^{μ}_{ρ} h^{ρ ν} - a (h) h^{μν} + b (h) η^{μν} .$
VII-A-ST@1#eq:density-expansion · line 475
$ρ_{γ} A_{γ}^{μν} = 1 + κa (h) + κ^{2} (a (k) + b (h)) (mod κ^{3}), = η^{μν} + κ A^{μν} (h) + κ^{2} (A^{μν} (k) + B^{μν} (h)) (mod κ^{3}) .$
unlabeled · line 470
$ρ_{γ} = - det γ, A_{γ}^{μν} = ρ_{γ} γ^{μν},$
unlabeled · line 486
$γ^{- 1} = η^{- 1} - κ h^{♯} + κ^{2} (h^{♯} h^{♯} - k^{♯}),$

Switching before variation

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