Manuscript Em02-D@4 · Em02-D-v4.tex · section · fcf2fac7

Characteristic chart

Fixed model, root, and localization › Characteristic chart · lines 364–425

Em02-D@4#eq:chart, Em02-D@4#eq:zeta, Em02-D@4#eq:local-data

Near $n_{*} = (1, 0, 0)$ , choose a smooth orthonormal spatial tangent frame $e_{A} (n)$ , and put

ℓ_{n} = (1, n), k_{n} = (1, - n), E_{A} = (0, e_{A}) .

Use

z b y = a + r, = a + β ℓ_{n} + u k_{n}, = a + α ℓ_{n} + v k_{n} + ξ^{A} E_{A} .

The inverse includes

n = \frac{b - a}{∣ b - a ∣}, β = \frac{b ^{0} - a ^{0} + ∣ b - a ∣}{2}, u = \frac{b ^{0} - a ^{0} - ∣ b - a ∣}{2} .

The other variables are the longitudinal and transverse projections of $y - a$ . The exact cone equations are

S^{2} = 4 β u, x^{2} = 4 α v - ∣ ξ ∣^{2}, (S - x)^{2} = 4 (β - α) (u - v) - ∣ ξ ∣^{2} .

Define

N = {r = 0, u = v = 0, ξ = 0} .

Fix $h = 1/100$ and

ζ = ϑ (∣ a ∣_{E}^{2} / h^{2}) ϑ (∣ n - n_{*} ∣^{2} / h^{2}) ϑ ((α - 1)^{2} / h^{2}) ϑ ((β - 2)^{2} / h^{2}) \times ϑ ((u^{2} + v^{2} + ∣ ξ ∣^{2}) / h^{2}) ϑ (∣ r ∣_{E}^{2} / h^{2}) .

Extend it by zero outside the chart. Its support is compact in a region with positive lower bounds on $α, β - α, β$ . On a neighborhood of its support,

f (b) = F (b) = 1 + b^{0}, \partial_{0} f (b) = \partial_{0} F (b) = 1, g (b) = g (y) = 1.

The outer propagation $S = b - a$ never reaches its vertex.

Displayed formulas in this passage

Em02-D@4#eq:chart · line 374
$z b y = a + r, = a + β ℓ_{n} + u k_{n}, = a + α ℓ_{n} + v k_{n} + ξ^{A} E_{A} .$
Em02-D@4#eq:zeta · line 404
$ζ = ϑ (∣ a ∣_{E}^{2} / h^{2}) ϑ (∣ n - n_{*} ∣^{2} / h^{2}) ϑ ((α - 1)^{2} / h^{2}) ϑ ((β - 2)^{2} / h^{2}) \times ϑ ((u^{2} + v^{2} + ∣ ξ ∣^{2}) / h^{2}) ϑ (∣ r ∣_{E}^{2} / h^{2}) .$
Em02-D@4#eq:local-data · line 418
$f (b) = F (b) = 1 + b^{0}, \partial_{0} f (b) = \partial_{0} F (b) = 1, g (b) = g (y) = 1.$
unlabeled · line 368
$ℓ_{n} = (1, n), k_{n} = (1, - n), E_{A} = (0, e_{A}) .$
unlabeled · line 384
$n = \frac{b - a}{∣ b - a ∣}, β = \frac{b ^{0} - a ^{0} + ∣ b - a ∣}{2}, u = \frac{b ^{0} - a ^{0} - ∣ b - a ∣}{2} .$
unlabeled · line 393
$S^{2} = 4 β u, x^{2} = 4 α v - ∣ ξ ∣^{2}, (S - x)^{2} = 4 (β - α) (u - v) - ∣ ξ ∣^{2} .$
unlabeled · line 399
$N = {r = 0, u = v = 0, ξ = 0} .$