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

Formal metric and density coefficients

The specified switched scalar action › Formal metric and density coefficients · lines 456–521

VII-A@1#eq:metricjet, VII-A@1#eq:ab, VII-A@1#eq:A, VII-A@1#eq:B, VII-A@1#eq:rhoexp, VII-A@1#eq:Adensityexp

Work coefficientwise in

C^{\infty} (M_{+}) [[κ]] / (κ^{3}) .

No convergence of the formal series or global invertibility of a perturbed coordinate system is assumed.

Let

g_{μν} = η_{μν} + κ h_{μν} + κ^{2} k_{μν} + O (κ^{3}),

where $h$ and $k$ are smooth symmetric tensors. Put

τ (h) = η^{μν} h_{μν}, h^{2} = h_{μν} h^{μν},

and define

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

Writing

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

one obtains

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

Displayed formulas in this passage

VII-A@1#eq:metricjet · line 466
$g_{μν} = η_{μν} + κ h_{μν} + κ^{2} k_{μν} + O (κ^{3}),$
VII-A@1#eq:ab · line 480
$a (h) A^{μν} (h) B^{μν} (h) = \frac{1}{2} τ (h), = \frac{1}{2} τ (h) η^{μν} - h^{μν}, = h^{μ}_{ρ} h^{ρ ν} - \frac{1}{2} τ (h) h^{μν} + b (h) η^{μν} . b (h) = \frac{1}{8} τ (h)^{2} - \frac{1}{4} h^{2},$
VII-A@1#eq:A · line 480
$a (h) A^{μν} (h) B^{μν} (h) = \frac{1}{2} τ (h), = \frac{1}{2} τ (h) η^{μν} - h^{μν}, = h^{μ}_{ρ} h^{ρ ν} - \frac{1}{2} τ (h) h^{μν} + b (h) η^{μν} . b (h) = \frac{1}{8} τ (h)^{2} - \frac{1}{4} h^{2},$
VII-A@1#eq:B · line 480
$a (h) A^{μν} (h) B^{μν} (h) = \frac{1}{2} τ (h), = \frac{1}{2} τ (h) η^{μν} - h^{μν}, = h^{μ}_{ρ} h^{ρ ν} - \frac{1}{2} τ (h) h^{μν} + b (h) η^{μν} . b (h) = \frac{1}{8} τ (h)^{2} - \frac{1}{4} h^{2},$
VII-A@1#eq:rhoexp · line 506
$ρ_{g} A_{g}^{μν} = 1 + κa (h) + κ^{2} (a (k) + b (h)) + O (κ^{3}), = η^{μν} + κ A^{μν} (h) + κ^{2} (A^{μν} (k) + B^{μν} (h)) + O (κ^{3}) .$
VII-A@1#eq:Adensityexp · line 506
$ρ_{g} A_{g}^{μν} = 1 + κa (h) + κ^{2} (a (k) + b (h)) + O (κ^{3}), = η^{μν} + κ A^{μν} (h) + κ^{2} (A^{μν} (k) + B^{μν} (h)) + O (κ^{3}) .$
unlabeled · line 459
$C^{\infty} (M_{+}) [[κ]] / (κ^{3}) .$
unlabeled · line 474
$τ (h) = η^{μν} h_{μν}, h^{2} = h_{μν} h^{μν},$
unlabeled · line 499
$ρ_{g} = - det g, A_{g}^{μν} = ρ_{g} g^{μν},$