Manuscript XIX-A@1 · XIX-A.tex · section · e30d8062

An explicit comparison prescription

The finite-time harmonic relational scalar › An explicit comparison prescription · lines 266–306

XIX-A@1#eq:metric-jet, XIX-A@1#eq:density-jet, XIX-A@1#eq:A1, XIX-A@1#eq:A2

Let

M = [t_{0}, t_{1}] \times R^{3}, η = - d t^{2} + δ_{ij} d x^{i} d x^{j} .

All fields extend smoothly to a neighborhood of the closed slab. We use a formal parameter $e$ and work modulo $e^{3}$ :

q_{e} = η + e h + \frac{e ^{2}}{2} k .

The coefficients $h, k$ are real smooth symmetric tensors with spatial supports contained in one fixed compact set over the slab. Indices in this subsection are raised with $η$ .

Write the inverse metric density as

a_{e}^{μν} := - det q_{e} q_{e}^{μν} = η^{μν} + e A_{1}^{μν} + \frac{e ^{2}}{2} A_{2}^{μν} .

Put $h_{tr} = η^{μν} h_{μν}$ and $k_{tr} = η^{μν} k_{μν}$ . Direct matrix expansion gives

A_{1}^{μν} A_{2}^{μν} = \frac{1}{2} h_{tr} η^{μν} - h^{μν}, = \frac{1}{2} k_{tr} η^{μν} - k^{μν} + 2 h^{μ}_{ρ} h^{ρ ν} - h_{tr} h^{μν} + (\frac{1}{4} h_{tr}^{2} - \frac{1}{2} h_{ρ σ} h^{ρ σ}) η^{μν} .

Displayed formulas in this passage

XIX-A@1#eq:metric-jet · line 275
$q_{e} = η + e h + \frac{e ^{2}}{2} k .$
XIX-A@1#eq:density-jet · line 284
$a_{e}^{μν} := - det q_{e} q_{e}^{μν} = η^{μν} + e A_{1}^{μν} + \frac{e ^{2}}{2} A_{2}^{μν} .$
XIX-A@1#eq:A1 · line 293
$A_{1}^{μν} A_{2}^{μν} = \frac{1}{2} h_{tr} η^{μν} - h^{μν}, = \frac{1}{2} k_{tr} η^{μν} - k^{μν} + 2 h^{μ}_{ρ} h^{ρ ν} - h_{tr} h^{μν} + (\frac{1}{4} h_{tr}^{2} - \frac{1}{2} h_{ρ σ} h^{ρ σ}) η^{μν} .$
XIX-A@1#eq:A2 · line 293
$A_{1}^{μν} A_{2}^{μν} = \frac{1}{2} h_{tr} η^{μν} - h^{μν}, = \frac{1}{2} k_{tr} η^{μν} - k^{μν} + 2 h^{μ}_{ρ} h^{ρ ν} - h_{tr} h^{μν} + (\frac{1}{4} h_{tr}^{2} - \frac{1}{2} h_{ρ σ} h^{ρ σ}) η^{μν} .$
unlabeled · line 269
$M = [t_{0}, t_{1}] \times R^{3}, η = - d t^{2} + δ_{ij} d x^{i} d x^{j} .$

An explicit comparison prescription

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