Manuscript X-A-conv-ST@1 · X-A-conv-ST.tex · section · 8ca47088

Auxiliary coordinates through second order

Stage B: Cauchy solution and regulator preflight › Auxiliary coordinates through second order · lines 531–572

X-A-conv-ST@1#eq:a1, X-A-conv-ST@1#eq:a2, X-A-conv-ST@1#eq:C, X-A-conv-ST@1#eq:harmonic

For a symmetric tensor $A$ , set

a_{1} (A)^{μν} a_{2} (A)^{μν} C^{μ} A = \frac{1}{2} (tr A) η^{μν} - A^{μν}, = A^{μ}_{ρ} A^{ρ ν} - \frac{1}{2} (tr A) A^{μν} + (\frac{1}{8} (tr A)^{2} - \frac{1}{4} A_{ρ σ} A^{ρ σ}) η^{μν}, = \partial_{ν} A^{ν μ} - \frac{1}{2} \partial^{μ} tr A = - \partial_{ν} a_{1} (A)^{ν μ} .

Thus

- det q q^{α β} = η^{α β} + κ a_{1} (H)^{α β} + κ^{2} (a_{1} (J)^{α β} + a_{2} (H)^{α β}) (mod κ^{3}) .

The scalar coordinate functions, with external label $μ$ , are defined by

\partial_{α} (- det q q^{α β} \partial_{β} X^{μ}) = 0, X^{μ} = x^{μ} + κ w^{μ} + κ^{2} v^{μ},

with the coordinate Cauchy data of $x^{μ}$ .

Displayed formulas in this passage

X-A-conv-ST@1#eq:a1 · line 534
$a_{1} (A)^{μν} a_{2} (A)^{μν} C^{μ} A = \frac{1}{2} (tr A) η^{μν} - A^{μν}, = A^{μ}_{ρ} A^{ρ ν} - \frac{1}{2} (tr A) A^{μν} + (\frac{1}{8} (tr A)^{2} - \frac{1}{4} A_{ρ σ} A^{ρ σ}) η^{μν}, = \partial_{ν} A^{ν μ} - \frac{1}{2} \partial^{μ} tr A = - \partial_{ν} a_{1} (A)^{ν μ} .$
X-A-conv-ST@1#eq:a2 · line 534
$a_{1} (A)^{μν} a_{2} (A)^{μν} C^{μ} A = \frac{1}{2} (tr A) η^{μν} - A^{μν}, = A^{μ}_{ρ} A^{ρ ν} - \frac{1}{2} (tr A) A^{μν} + (\frac{1}{8} (tr A)^{2} - \frac{1}{4} A_{ρ σ} A^{ρ σ}) η^{μν}, = \partial_{ν} A^{ν μ} - \frac{1}{2} \partial^{μ} tr A = - \partial_{ν} a_{1} (A)^{ν μ} .$
X-A-conv-ST@1#eq:C · line 534
$a_{1} (A)^{μν} a_{2} (A)^{μν} C^{μ} A = \frac{1}{2} (tr A) η^{μν} - A^{μν}, = A^{μ}_{ρ} A^{ρ ν} - \frac{1}{2} (tr A) A^{μν} + (\frac{1}{8} (tr A)^{2} - \frac{1}{4} A_{ρ σ} A^{ρ σ}) η^{μν}, = \partial_{ν} A^{ν μ} - \frac{1}{2} \partial^{μ} tr A = - \partial_{ν} a_{1} (A)^{ν μ} .$
X-A-conv-ST@1#eq:harmonic · line 563
$\partial_{α} (- det q q^{α β} \partial_{β} X^{μ}) = 0, X^{μ} = x^{μ} + κ w^{μ} + κ^{2} v^{μ},$
unlabeled · line 552
$- det q q^{α β} = η^{α β} + κ a_{1} (H)^{α β} + κ^{2} (a_{1} (J)^{α β} + a_{2} (H)^{α β}) (mod κ^{3}) .$