Manuscript X-D@3 · X-D-v3.tex · section · b7774529

Candidate coordinates determined by the layer

Propagation and convergence › Candidate coordinates determined by the layer · lines 866–904

X-D@3#eq:limit-cauchy, X-D@3#eq:delta-data

Define the right-hand limiting coordinates by

□ X_{1, 0}^{ρ} □ X_{2, 0}^{ρ} = - \partial_{μ} a^{μ ρ}, = - \partial_{μ} (A^{μ ρ} + a^{μν} \partial_{ν} X_{1, 0}^{ρ}), X_{1, 0}^{ρ} (0) X_{2, 0}^{ρ} (0) = 0, = 0, \partial_{t} X_{1, 0}^{ρ} (0) \partial_{t} X_{2, 0}^{ρ} (0) = v_{1}^{ρ}, = v_{2}^{ρ} .

These are ordinary smooth Cauchy problems on $t \geq 0$ , locally by finite propagation.

For an explicit statement of the data at $t = δ$ , let $s_{j}^{ρ}$ be the right-hand sources in (eq:limit-cauchy). Then

X_{j, 0}^{ρ} (δ) \partial_{t} X_{j, 0}^{ρ} (δ) = S (δ) v_{j}^{ρ} + \int_{0}^{δ} S (δ - t) s_{j}^{ρ} (t) d t, = C (δ) v_{j}^{ρ} + \int_{0}^{δ} C (δ - t) s_{j}^{ρ} (t) d t .

All layer contributions are included through the derived velocities.

Displayed formulas in this passage

X-D@3#eq:limit-cauchy · line 869
$□ X_{1, 0}^{ρ} □ X_{2, 0}^{ρ} = - \partial_{μ} a^{μ ρ}, = - \partial_{μ} (A^{μ ρ} + a^{μν} \partial_{ν} X_{1, 0}^{ρ}), X_{1, 0}^{ρ} (0) X_{2, 0}^{ρ} (0) = 0, = 0, \partial_{t} X_{1, 0}^{ρ} (0) \partial_{t} X_{2, 0}^{ρ} (0) = v_{1}^{ρ}, = v_{2}^{ρ} .$
X-D@3#eq:delta-data · line 892
$X_{j, 0}^{ρ} (δ) \partial_{t} X_{j, 0}^{ρ} (δ) = S (δ) v_{j}^{ρ} + \int_{0}^{δ} S (δ - t) s_{j}^{ρ} (t) d t, = C (δ) v_{j}^{ρ} + \int_{0}^{δ} C (δ - t) s_{j}^{ρ} (t) d t .$