Manuscript VI@1 · VI.tex · lemma · b5b60afb

Lemma: Second variation of the scalar wave operator

Finite-time harmonic coordinates and second-order descent › Explicit scalar operator expansion · lines 853–906 · proof included or linked

VI@1#lem:second-wave, VI@1#eq:M

The operator in (eq:wave-expansion) is

M_{h} u = \partial_{μ} (C_{h}^{μν} \partial_{ν} u) - d_{h} L_{h} u - e_{h} □_{η} u .

Proof. The inverse metric and volume density are

γ^{μν} - γ = η^{μν} - κ h^{μν} + κ^{2} (h^{μ}_{ρ} h^{ρ ν} - j^{μν}) mod κ^{3}, = 1 + κ d_{h} + κ^{2} (d_{j} + e_{h}) mod κ^{3} .

Hence

- γ γ^{μν} = η^{μν} + κ A_{h}^{μν} + κ^{2} (A_{j}^{μν} + C_{h}^{μν}) mod κ^{3} .

Substitute this and

(- γ)^{- 1} = 1 - κ d_{h} + κ^{2} (d_{h}^{2} - d_{j} - e_{h}) mod κ^{3}

into the divergence formula for $□_{γ}$ . The first coefficient is

\partial_{μ} (A_{h}^{μν} \partial_{ν}) - d_{h} □_{η} = L_{h} .

The part linear in $j$ is $L_{j}$ . The remaining part is

\partial_{μ} (C_{h}^{μν} \partial_{ν}) - d_{h} \partial_{μ} (A_{h}^{μν} \partial_{ν}) + (d_{h}^{2} - e_{h}) □_{η},

which equals (eq:M).

Displayed formulas in this passage

VI@1#eq:M · line 856
$M_{h} u = \partial_{μ} (C_{h}^{μν} \partial_{ν} u) - d_{h} L_{h} u - e_{h} □_{η} u .$
unlabeled · line 867
$γ^{μν} - γ = η^{μν} - κ h^{μν} + κ^{2} (h^{μ}_{ρ} h^{ρ ν} - j^{μν}) mod κ^{3}, = 1 + κ d_{h} + κ^{2} (d_{j} + e_{h}) mod κ^{3} .$
unlabeled · line 879
$- γ γ^{μν} = η^{μν} + κ A_{h}^{μν} + κ^{2} (A_{j}^{μν} + C_{h}^{μν}) mod κ^{3} .$
unlabeled · line 888
$(- γ)^{- 1} = 1 - κ d_{h} + κ^{2} (d_{h}^{2} - d_{j} - e_{h}) mod κ^{3}$
unlabeled · line 896
$\partial_{μ} (A_{h}^{μν} \partial_{ν}) - d_{h} □_{η} = L_{h} .$
unlabeled · line 900
$\partial_{μ} (C_{h}^{μν} \partial_{ν}) - d_{h} \partial_{μ} (A_{h}^{μν} \partial_{ν}) + (d_{h}^{2} - e_{h}) □_{η},$