Manuscript VII-A@1 · VII-A.tex · lemma · 11da3fe6

Lemma: Density expansion

The specified switched scalar action › Formal metric and density coefficients · lines 522–541 · proof included or linked

Equations (eq:rhoexp)–(eq:Adensityexp) follow from (eq:metricjet).

Proof. The inverse metric is

g^{μν} = η^{μν} - κ h^{μν} + κ^{2} (h^{μ}_{ρ} h^{ρ ν} - k^{μν}) + O (κ^{3}) .

Expanding $exp (\frac{1}{2} tr lo g (id + κ η^{- 1} h + κ^{2} η^{- 1} k))$ gives (eq:rhoexp). Multiplication gives (eq:Adensityexp).

Displayed formulas in this passage

unlabeled · line 529
$g^{μν} = η^{μν} - κ h^{μν} + κ^{2} (h^{μ}_{ρ} h^{ρ ν} - k^{μν}) + O (κ^{3}) .$