Manuscript X-F-ST@1 · X-F-ST.tex · section · e71f03ef

Density coefficients and coordinate prescription

Frozen analytical data and the scalar insertion target › Density coefficients and coordinate prescription · lines 406–456

X-F-ST@1#eq:coefficients, X-F-ST@1#eq:coordinates, X-F-ST@1#eq:L

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Indices without an indicated metric are moved by $η$ . For $H = tr_{η} l$ , define

a_{1}^{μν} [l] b_{2} [l] a_{2}^{μν} [l] A_{2} [h, k] = \frac{1}{2} H η^{μν} - l^{μν}, = \frac{1}{8} H^{2} - \frac{1}{4} l_{α β} l^{α β}, = l^{μ}_{λ} l^{λ ν} - \frac{1}{2} H l^{μν} + b_{2} [l] η^{μν}, = a_{1} [k] + a_{2} [h], b_{1} [l] B_{2} [h, k] = \frac{1}{2} H, = b_{1} [k] + b_{2} [h] .

These are the coefficients of $J_{q} q^{μν}$ and $J_{q} = - det q$ for $q = η + κh + κ^{2} k$ .

The zero-data coordinate coefficients are

□ w^{ρ} [l] □ v^{ρ} [h, k] = - \partial_{μ} (σ a_{1}^{μ ρ} [l]), = - \partial_{μ} (σ {A_{2}^{μ ρ} [h, k] + a_{1}^{μν} [h] \partial_{ν} w^{ρ} [h]}) .

Their positions and velocities vanish at $Σ$ . For nondecaying data these equations are defined locally by finite-time Cauchy theory and finite propagation, not by an unjustified integral over an infinite past.

The scalar Euler coefficients are

L_{1} [l] u L_{2} [h, k] u = \partial_{μ} (g a_{1}^{μν} [l] \partial_{ν} u) + g m^{2} b_{1} [l] u, = \partial_{μ} (g A_{2}^{μν} [h, k] \partial_{ν} u) + g m^{2} B_{2} [h, k] u .

Every switch derivative remains inside these divergences.

Displayed formulas in this passage

X-F-ST@1#eq:coefficients · line 410
$a_{1}^{μν} [l] b_{2} [l] a_{2}^{μν} [l] A_{2} [h, k] = \frac{1}{2} H η^{μν} - l^{μν}, = \frac{1}{8} H^{2} - \frac{1}{4} l_{α β} l^{α β}, = l^{μ}_{λ} l^{λ ν} - \frac{1}{2} H l^{μν} + b_{2} [l] η^{μν}, = a_{1} [k] + a_{2} [h], b_{1} [l] B_{2} [h, k] = \frac{1}{2} H, = b_{1} [k] + b_{2} [h] .$
X-F-ST@1#eq:coordinates · line 429
$□ w^{ρ} [l] □ v^{ρ} [h, k] = - \partial_{μ} (σ a_{1}^{μ ρ} [l]), = - \partial_{μ} (σ {A_{2}^{μ ρ} [h, k] + a_{1}^{μν} [h] \partial_{ν} w^{ρ} [h]}) .$
X-F-ST@1#eq:L · line 445
$L_{1} [l] u L_{2} [h, k] u = \partial_{μ} (g a_{1}^{μν} [l] \partial_{ν} u) + g m^{2} b_{1} [l] u, = \partial_{μ} (g A_{2}^{μν} [h, k] \partial_{ν} u) + g m^{2} B_{2} [h, k] u .$