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

Definition and boundary prescription

Unswitched harmonic coordinates › Definition and boundary prescription · lines 726–781

VII-A@1#eq:harmonic, VII-A@1#eq:harmonicdata, VII-A@1#eq:Xexp, VII-A@1#eq:C, VII-A@1#eq:w, VII-A@1#eq:v

The four functions $X_{g}^{μ}$ are defined coefficientwise by

\partial_{α} (A_{g}^{α β} \partial_{β} X_{g}^{μ}) = 0,

with Cauchy data

X_{g}^{μ} ∣_{Σ_{0}} = x^{μ} ∣_{Σ_{0}}, \partial_{t} X_{g}^{μ} ∣_{Σ_{0}} = \partial_{t} x^{μ} .

There is no switch in (eq:harmonic).

Write

X_{g}^{μ} = x^{μ} + κ w^{μ} [h] + κ^{2} v^{μ} [h, k] + O (κ^{3}) .

Define

C^{μ} [h] = \partial_{ν} h^{ν μ} - \frac{1}{2} \partial^{μ} τ (h) = - \partial_{ν} A^{ν μ} (h) .

Then

w^{μ} [h] v^{μ} [h, k] = E_{0} C^{μ} [h], = E_{0} C^{μ} [k] - E_{0} (\partial_{α} B^{α μ} (h) + Q_{h}^{(0)} w^{μ} [h]) .

Displayed formulas in this passage

VII-A@1#eq:harmonic · line 730
$\partial_{α} (A_{g}^{α β} \partial_{β} X_{g}^{μ}) = 0,$
VII-A@1#eq:harmonicdata · line 738
$X_{g}^{μ} ∣_{Σ_{0}} = x^{μ} ∣_{Σ_{0}}, \partial_{t} X_{g}^{μ} ∣_{Σ_{0}} = \partial_{t} x^{μ} .$
VII-A@1#eq:Xexp · line 748
$X_{g}^{μ} = x^{μ} + κ w^{μ} [h] + κ^{2} v^{μ} [h, k] + O (κ^{3}) .$
VII-A@1#eq:C · line 756
$C^{μ} [h] = \partial_{ν} h^{ν μ} - \frac{1}{2} \partial^{μ} τ (h) = - \partial_{ν} A^{ν μ} (h) .$
VII-A@1#eq:w · line 766
$w^{μ} [h] v^{μ} [h, k] = E_{0} C^{μ} [h], = E_{0} C^{μ} [k] - E_{0} (\partial_{α} B^{α μ} (h) + Q_{h}^{(0)} w^{μ} [h]) .$
VII-A@1#eq:v · line 766
$w^{μ} [h] v^{μ} [h, k] = E_{0} C^{μ} [h], = E_{0} C^{μ} [k] - E_{0} (\partial_{α} B^{α μ} (h) + Q_{h}^{(0)} w^{μ} [h]) .$